LMFDB - L-function 40-160e20-1.1-c5e20-0-0

Dirichlet series

L(s) = 1

+ 196·7^-s + 1.62e3·9^-s + 4.67e3·23^-s − 6.25e3·25^-s − 7.16e3·31^-s + 1.16e4·41^-s − 4.41e4·47^-s − 1.39e5·49^-s + 3.17e5·63^-s + 2.00e5·71^-s − 1.05e5·73^-s − 2.82e5·79^-s + 1.23e6·81^-s − 3.16e3·89^-s + 1.47e5·97^-s + 4.55e5·103^-s − 3.02e5·113^-s + 1.51e6·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 9.16e5·161^-s + 163^-s + ⋯

L(s) = 1

+ 1.51·7^-s + 20/3·9^-s + 1.84·23^-s − 2·25^-s − 1.33·31^-s + 1.07·41^-s − 2.91·47^-s − 8.29·49^-s + 10.0·63^-s + 4.71·71^-s − 2.30·73^-s − 5.08·79^-s + 20.8·81^-s − 0.0422·89^-s + 1.59·97^-s + 4.23·103^-s − 2.22·113^-s + 9.37·121^-s + 5.50e−6·127^-s + 5.09e−6·131^-s + 4.55e−6·137^-s + 4.38e−6·139^-s + 3.69e−6·149^-s + 3.56e−6·151^-s + 3.23e−6·157^-s + 2.78·161^-s + 2.94e−6·163^-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree:	$40$
Conductor:	$2^{100} \cdot 5^{20}$
Sign:	$1$
Analytic conductor:	$1.53324\times 10^{28}$
Root analytic conductor:	$5.06570$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{100} \cdot 5^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$34.47725220$
$L(\frac12)$	$\approx$	$34.47725220$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}

	$p$	$F_p(T)$
bad	2	$1$
	5	$( 1 + p^{4} T^{2} )^{10}$
good	3	$1 - 20 p^{4} T^{2} + 464774 p T^{4} - 834927892 T^{6} + 43647016469 p^{2} T^{8} - 51765303184624 p T^{10} + 53814329151823576 T^{12} - 1863599575911640208 p^{2} T^{14} + 19706806085941150310 p^{5} T^{16} -$ $17\!\cdots\!00$ $p^{6} T^{18} +$ $48\!\cdots\!00$ $p^{8} T^{20} -$ $17\!\cdots\!00$ $p^{16} T^{22} + 19706806085941150310 p^{25} T^{24} - 1863599575911640208 p^{32} T^{26} + 53814329151823576 p^{40} T^{28} - 51765303184624 p^{51} T^{30} + 43647016469 p^{62} T^{32} - 834927892 p^{70} T^{34} + 464774 p^{81} T^{36} - 20 p^{94} T^{38} + p^{100} T^{40}$
	7	$( 1 - 2 p^{2} T + 84148 T^{2} - 8017214 T^{3} + 3556570901 T^{4} - 345044190776 T^{5} + 103920201897616 T^{6} - 10159390994080936 T^{7} + 2363994840482709802 T^{8} -$ $22\!\cdots\!76$ $T^{9} +$ $43\!\cdots\!72$ $T^{10} -$ $22\!\cdots\!76$ $p^{5} T^{11} + 2363994840482709802 p^{10} T^{12} - 10159390994080936 p^{15} T^{13} + 103920201897616 p^{20} T^{14} - 345044190776 p^{25} T^{15} + 3556570901 p^{30} T^{16} - 8017214 p^{35} T^{17} + 84148 p^{40} T^{18} - 2 p^{47} T^{19} + p^{50} T^{20} )^{2}$
	11	$1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} +$ $22\!\cdots\!17$ $T^{8} -$ $70\!\cdots\!04$ $T^{10} +$ $18\!\cdots\!36$ $T^{12} -$ $40\!\cdots\!64$ $T^{14} +$ $79\!\cdots\!82$ $T^{16} -$ $14\!\cdots\!04$ $T^{18} +$ $23\!\cdots\!76$ $T^{20} -$ $14\!\cdots\!04$ $p^{10} T^{22} +$ $79\!\cdots\!82$ $p^{20} T^{24} -$ $40\!\cdots\!64$ $p^{30} T^{26} +$ $18\!\cdots\!36$ $p^{40} T^{28} -$ $70\!\cdots\!04$ $p^{50} T^{30} +$ $22\!\cdots\!17$ $p^{60} T^{32} - 592766540112799924 p^{70} T^{34} + 1155249727726 p^{80} T^{36} - 1510004 p^{90} T^{38} + p^{100} T^{40}$
	13	$1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} +$ $67\!\cdots\!57$ $T^{8} -$ $50\!\cdots\!12$ $T^{10} +$ $32\!\cdots\!64$ $T^{12} -$ $17\!\cdots\!28$ $T^{14} +$ $87\!\cdots\!42$ $T^{16} -$ $38\!\cdots\!32$ $T^{18} +$ $14\!\cdots\!64$ $T^{20} -$ $38\!\cdots\!32$ $p^{10} T^{22} +$ $87\!\cdots\!42$ $p^{20} T^{24} -$ $17\!\cdots\!28$ $p^{30} T^{26} +$ $32\!\cdots\!64$ $p^{40} T^{28} -$ $50\!\cdots\!12$ $p^{50} T^{30} +$ $67\!\cdots\!57$ $p^{60} T^{32} - 7441447313525468748 p^{70} T^{34} + 6227853839054 p^{80} T^{36} - 3520332 p^{90} T^{38} + p^{100} T^{40}$
	17	$( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} +$ $14\!\cdots\!00$ $T^{7} +$ $56\!\cdots\!10$ $T^{8} +$ $21\!\cdots\!00$ $T^{9} +$ $84\!\cdots\!72$ $T^{10} +$ $21\!\cdots\!00$ $p^{5} T^{11} +$ $56\!\cdots\!10$ $p^{10} T^{12} +$ $14\!\cdots\!00$ $p^{15} T^{13} + 32518332783929091752 p^{20} T^{14} + 7413542230528000 p^{25} T^{15} + 15317572824845 p^{30} T^{16} + 1859072000 p^{35} T^{17} + 5447662 p^{40} T^{18} + p^{50} T^{20} )^{2}$
	19	$1 - 23638692 T^{2} + 296336418074734 T^{4} -$ $25\!\cdots\!32$ $T^{6} +$ $17\!\cdots\!97$ $T^{8} -$ $49\!\cdots\!20$ $p T^{10} +$ $42\!\cdots\!52$ $T^{12} -$ $16\!\cdots\!48$ $T^{14} +$ $56\!\cdots\!66$ $T^{16} -$ $16\!\cdots\!12$ $T^{18} +$ $44\!\cdots\!28$ $T^{20} -$ $16\!\cdots\!12$ $p^{10} T^{22} +$ $56\!\cdots\!66$ $p^{20} T^{24} -$ $16\!\cdots\!48$ $p^{30} T^{26} +$ $42\!\cdots\!52$ $p^{40} T^{28} -$ $49\!\cdots\!20$ $p^{51} T^{30} +$ $17\!\cdots\!97$ $p^{60} T^{32} -$ $25\!\cdots\!32$ $p^{70} T^{34} + 296336418074734 p^{80} T^{36} - 23638692 p^{90} T^{38} + p^{100} T^{40}$
	23	$( 1 - 2338 T + 35997660 T^{2} - 45007042654 T^{3} + 520972471777845 T^{4} - 50426407997609208 T^{5} +$ $39\!\cdots\!60$ $T^{6} +$ $66\!\cdots\!96$ $T^{7} +$ $17\!\cdots\!10$ $T^{8} +$ $91\!\cdots\!52$ $T^{9} +$ $76\!\cdots\!60$ $T^{10} +$ $91\!\cdots\!52$ $p^{5} T^{11} +$ $17\!\cdots\!10$ $p^{10} T^{12} +$ $66\!\cdots\!96$ $p^{15} T^{13} +$ $39\!\cdots\!60$ $p^{20} T^{14} - 50426407997609208 p^{25} T^{15} + 520972471777845 p^{30} T^{16} - 45007042654 p^{35} T^{17} + 35997660 p^{40} T^{18} - 2338 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	29	$1 - 215092900 T^{2} + 23243889276296494 T^{4} -$ $16\!\cdots\!00$ $T^{6} +$ $90\!\cdots\!13$ $T^{8} -$ $38\!\cdots\!00$ $T^{10} +$ $13\!\cdots\!92$ $T^{12} -$ $42\!\cdots\!00$ $T^{14} +$ $11\!\cdots\!86$ $T^{16} -$ $27\!\cdots\!00$ $T^{18} +$ $58\!\cdots\!28$ $T^{20} -$ $27\!\cdots\!00$ $p^{10} T^{22} +$ $11\!\cdots\!86$ $p^{20} T^{24} -$ $42\!\cdots\!00$ $p^{30} T^{26} +$ $13\!\cdots\!92$ $p^{40} T^{28} -$ $38\!\cdots\!00$ $p^{50} T^{30} +$ $90\!\cdots\!13$ $p^{60} T^{32} -$ $16\!\cdots\!00$ $p^{70} T^{34} + 23243889276296494 p^{80} T^{36} - 215092900 p^{90} T^{38} + p^{100} T^{40}$
	31	$( 1 + 3580 T + 132421614 T^{2} + 484936307876 T^{3} + 8983138546835629 T^{4} + 33755686429634218608 T^{5} +$ $43\!\cdots\!88$ $T^{6} +$ $16\!\cdots\!96$ $T^{7} +$ $17\!\cdots\!38$ $T^{8} +$ $60\!\cdots\!32$ $T^{9} +$ $54\!\cdots\!44$ $T^{10} +$ $60\!\cdots\!32$ $p^{5} T^{11} +$ $17\!\cdots\!38$ $p^{10} T^{12} +$ $16\!\cdots\!96$ $p^{15} T^{13} +$ $43\!\cdots\!88$ $p^{20} T^{14} + 33755686429634218608 p^{25} T^{15} + 8983138546835629 p^{30} T^{16} + 484936307876 p^{35} T^{17} + 132421614 p^{40} T^{18} + 3580 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	37	$1 - 565372668 T^{2} + 171401952644913934 T^{4} -$ $36\!\cdots\!00$ $T^{6} +$ $59\!\cdots\!09$ $T^{8} -$ $80\!\cdots\!44$ $T^{10} +$ $93\!\cdots\!16$ $T^{12} -$ $93\!\cdots\!40$ $T^{14} +$ $83\!\cdots\!86$ $T^{16} -$ $67\!\cdots\!48$ $T^{18} +$ $48\!\cdots\!08$ $T^{20} -$ $67\!\cdots\!48$ $p^{10} T^{22} +$ $83\!\cdots\!86$ $p^{20} T^{24} -$ $93\!\cdots\!40$ $p^{30} T^{26} +$ $93\!\cdots\!16$ $p^{40} T^{28} -$ $80\!\cdots\!44$ $p^{50} T^{30} +$ $59\!\cdots\!09$ $p^{60} T^{32} -$ $36\!\cdots\!00$ $p^{70} T^{34} + 171401952644913934 p^{80} T^{36} - 565372668 p^{90} T^{38} + p^{100} T^{40}$
	41	$( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} -$ $13\!\cdots\!48$ $T^{5} +$ $39\!\cdots\!40$ $T^{6} -$ $28\!\cdots\!68$ $T^{7} +$ $63\!\cdots\!42$ $T^{8} -$ $43\!\cdots\!24$ $T^{9} +$ $82\!\cdots\!24$ $T^{10} -$ $43\!\cdots\!24$ $p^{5} T^{11} +$ $63\!\cdots\!42$ $p^{10} T^{12} -$ $28\!\cdots\!68$ $p^{15} T^{13} +$ $39\!\cdots\!40$ $p^{20} T^{14} -$ $13\!\cdots\!48$ $p^{25} T^{15} + 181136735899530493 p^{30} T^{16} - 4176614475628 p^{35} T^{17} + 580737234 p^{40} T^{18} - 5804 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	43	$1 - 1208126740 T^{2} + 787740581508660018 T^{4} -$ $36\!\cdots\!96$ $T^{6} +$ $12\!\cdots\!73$ $T^{8} -$ $37\!\cdots\!80$ $T^{10} +$ $95\!\cdots\!40$ $T^{12} -$ $20\!\cdots\!56$ $T^{14} +$ $40\!\cdots\!70$ $T^{16} -$ $70\!\cdots\!00$ $T^{18} +$ $10\!\cdots\!96$ $T^{20} -$ $70\!\cdots\!00$ $p^{10} T^{22} +$ $40\!\cdots\!70$ $p^{20} T^{24} -$ $20\!\cdots\!56$ $p^{30} T^{26} +$ $95\!\cdots\!40$ $p^{40} T^{28} -$ $37\!\cdots\!80$ $p^{50} T^{30} +$ $12\!\cdots\!73$ $p^{60} T^{32} -$ $36\!\cdots\!96$ $p^{70} T^{34} + 787740581508660018 p^{80} T^{36} - 1208126740 p^{90} T^{38} + p^{100} T^{40}$
	47	$( 1 + 10 p^{2} T + 1548510076 T^{2} + 25779450599270 T^{3} + 1065218725316011845 T^{4} +$ $14\!\cdots\!20$ $T^{5} +$ $45\!\cdots\!96$ $T^{6} +$ $51\!\cdots\!60$ $T^{7} +$ $14\!\cdots\!10$ $T^{8} +$ $30\!\cdots\!00$ $p T^{9} +$ $36\!\cdots\!56$ $T^{10} +$ $30\!\cdots\!00$ $p^{6} T^{11} +$ $14\!\cdots\!10$ $p^{10} T^{12} +$ $51\!\cdots\!60$ $p^{15} T^{13} +$ $45\!\cdots\!96$ $p^{20} T^{14} +$ $14\!\cdots\!20$ $p^{25} T^{15} + 1065218725316011845 p^{30} T^{16} + 25779450599270 p^{35} T^{17} + 1548510076 p^{40} T^{18} + 10 p^{47} T^{19} + p^{50} T^{20} )^{2}$
	53	$1 - 4003669356 T^{2} + 7954725909905649454 T^{4} -$ $10\!\cdots\!60$ $T^{6} +$ $10\!\cdots\!77$ $T^{8} -$ $91\!\cdots\!76$ $T^{10} +$ $65\!\cdots\!16$ $T^{12} -$ $40\!\cdots\!80$ $T^{14} +$ $22\!\cdots\!54$ $T^{16} -$ $11\!\cdots\!56$ $T^{18} +$ $48\!\cdots\!96$ $T^{20} -$ $11\!\cdots\!56$ $p^{10} T^{22} +$ $22\!\cdots\!54$ $p^{20} T^{24} -$ $40\!\cdots\!80$ $p^{30} T^{26} +$ $65\!\cdots\!16$ $p^{40} T^{28} -$ $91\!\cdots\!76$ $p^{50} T^{30} +$ $10\!\cdots\!77$ $p^{60} T^{32} -$ $10\!\cdots\!60$ $p^{70} T^{34} + 7954725909905649454 p^{80} T^{36} - 4003669356 p^{90} T^{38} + p^{100} T^{40}$
	59	$1 - 9996949828 T^{2} + 49650800837889885966 T^{4} -$ $16\!\cdots\!20$ $T^{6} +$ $39\!\cdots\!17$ $T^{8} -$ $73\!\cdots\!28$ $T^{10} +$ $11\!\cdots\!44$ $T^{12} -$ $14\!\cdots\!80$ $T^{14} +$ $15\!\cdots\!54$ $T^{16} -$ $13\!\cdots\!88$ $T^{18} +$ $10\!\cdots\!24$ $T^{20} -$ $13\!\cdots\!88$ $p^{10} T^{22} +$ $15\!\cdots\!54$ $p^{20} T^{24} -$ $14\!\cdots\!80$ $p^{30} T^{26} +$ $11\!\cdots\!44$ $p^{40} T^{28} -$ $73\!\cdots\!28$ $p^{50} T^{30} +$ $39\!\cdots\!17$ $p^{60} T^{32} -$ $16\!\cdots\!20$ $p^{70} T^{34} + 49650800837889885966 p^{80} T^{36} - 9996949828 p^{90} T^{38} + p^{100} T^{40}$
	61	$1 - 7152852348 T^{2} + 26523669582205677166 T^{4} -$ $67\!\cdots\!00$ $T^{6} +$ $13\!\cdots\!17$ $T^{8} -$ $22\!\cdots\!48$ $T^{10} +$ $31\!\cdots\!84$ $T^{12} -$ $38\!\cdots\!60$ $T^{14} +$ $42\!\cdots\!14$ $T^{16} -$ $41\!\cdots\!08$ $T^{18} +$ $36\!\cdots\!64$ $T^{20} -$ $41\!\cdots\!08$ $p^{10} T^{22} +$ $42\!\cdots\!14$ $p^{20} T^{24} -$ $38\!\cdots\!60$ $p^{30} T^{26} +$ $31\!\cdots\!84$ $p^{40} T^{28} -$ $22\!\cdots\!48$ $p^{50} T^{30} +$ $13\!\cdots\!17$ $p^{60} T^{32} -$ $67\!\cdots\!00$ $p^{70} T^{34} + 26523669582205677166 p^{80} T^{36} - 7152852348 p^{90} T^{38} + p^{100} T^{40}$
	67	$1 - 11828518964 T^{2} + 68669252417166303634 T^{4} -$ $26\!\cdots\!60$ $T^{6} +$ $76\!\cdots\!57$ $T^{8} -$ $18\!\cdots\!04$ $T^{10} +$ $38\!\cdots\!16$ $T^{12} -$ $70\!\cdots\!40$ $T^{14} +$ $11\!\cdots\!54$ $T^{16} -$ $18\!\cdots\!64$ $T^{18} +$ $25\!\cdots\!76$ $T^{20} -$ $18\!\cdots\!64$ $p^{10} T^{22} +$ $11\!\cdots\!54$ $p^{20} T^{24} -$ $70\!\cdots\!40$ $p^{30} T^{26} +$ $38\!\cdots\!16$ $p^{40} T^{28} -$ $18\!\cdots\!04$ $p^{50} T^{30} +$ $76\!\cdots\!57$ $p^{60} T^{32} -$ $26\!\cdots\!60$ $p^{70} T^{34} + 68669252417166303634 p^{80} T^{36} - 11828518964 p^{90} T^{38} + p^{100} T^{40}$
	71	$( 1 - 100156 T + 13448448446 T^{2} - 957445823975748 T^{3} + 76873855601451932317 T^{4} -$ $43\!\cdots\!20$ $T^{5} +$ $26\!\cdots\!28$ $T^{6} -$ $12\!\cdots\!92$ $T^{7} +$ $67\!\cdots\!74$ $T^{8} -$ $28\!\cdots\!84$ $T^{9} +$ $13\!\cdots\!68$ $T^{10} -$ $28\!\cdots\!84$ $p^{5} T^{11} +$ $67\!\cdots\!74$ $p^{10} T^{12} -$ $12\!\cdots\!92$ $p^{15} T^{13} +$ $26\!\cdots\!28$ $p^{20} T^{14} -$ $43\!\cdots\!20$ $p^{25} T^{15} + 76873855601451932317 p^{30} T^{16} - 957445823975748 p^{35} T^{17} + 13448448446 p^{40} T^{18} - 100156 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	73	$( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} +$ $11\!\cdots\!56$ $T^{5} +$ $82\!\cdots\!04$ $T^{6} +$ $19\!\cdots\!04$ $T^{7} +$ $11\!\cdots\!58$ $T^{8} +$ $24\!\cdots\!96$ $T^{9} +$ $16\!\cdots\!04$ $T^{10} +$ $24\!\cdots\!96$ $p^{5} T^{11} +$ $11\!\cdots\!58$ $p^{10} T^{12} +$ $19\!\cdots\!04$ $p^{15} T^{13} +$ $82\!\cdots\!04$ $p^{20} T^{14} +$ $11\!\cdots\!56$ $p^{25} T^{15} + 37721970904921608509 p^{30} T^{16} + 374528688123736 p^{35} T^{17} + 9379342894 p^{40} T^{18} + 52568 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	79	$( 1 + 141040 T + 30508439526 T^{2} + 3027236690742416 T^{3} +$ $38\!\cdots\!93$ $T^{4} +$ $29\!\cdots\!88$ $T^{5} +$ $27\!\cdots\!48$ $T^{6} +$ $17\!\cdots\!16$ $T^{7} +$ $13\!\cdots\!30$ $T^{8} +$ $92\!\cdots\!60$ $p T^{9} +$ $47\!\cdots\!04$ $T^{10} +$ $92\!\cdots\!60$ $p^{6} T^{11} +$ $13\!\cdots\!30$ $p^{10} T^{12} +$ $17\!\cdots\!16$ $p^{15} T^{13} +$ $27\!\cdots\!48$ $p^{20} T^{14} +$ $29\!\cdots\!88$ $p^{25} T^{15} +$ $38\!\cdots\!93$ $p^{30} T^{16} + 3027236690742416 p^{35} T^{17} + 30508439526 p^{40} T^{18} + 141040 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	83	$1 - 34768653380 T^{2} +$ $55\!\cdots\!58$ $T^{4} -$ $53\!\cdots\!56$ $T^{6} +$ $34\!\cdots\!33$ $T^{8} -$ $15\!\cdots\!20$ $T^{10} +$ $44\!\cdots\!40$ $T^{12} -$ $24\!\cdots\!36$ $T^{14} -$ $64\!\cdots\!10$ $T^{16} +$ $49\!\cdots\!20$ $T^{18} -$ $23\!\cdots\!44$ $T^{20} +$ $49\!\cdots\!20$ $p^{10} T^{22} -$ $64\!\cdots\!10$ $p^{20} T^{24} -$ $24\!\cdots\!36$ $p^{30} T^{26} +$ $44\!\cdots\!40$ $p^{40} T^{28} -$ $15\!\cdots\!20$ $p^{50} T^{30} +$ $34\!\cdots\!33$ $p^{60} T^{32} -$ $53\!\cdots\!56$ $p^{70} T^{34} +$ $55\!\cdots\!58$ $p^{80} T^{36} - 34768653380 p^{90} T^{38} + p^{100} T^{40}$
	89	$( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} +$ $38\!\cdots\!93$ $T^{4} +$ $90\!\cdots\!72$ $T^{5} +$ $38\!\cdots\!08$ $T^{6} +$ $95\!\cdots\!24$ $T^{7} +$ $29\!\cdots\!70$ $T^{8} +$ $73\!\cdots\!60$ $T^{9} +$ $18\!\cdots\!64$ $T^{10} +$ $73\!\cdots\!60$ $p^{5} T^{11} +$ $29\!\cdots\!70$ $p^{10} T^{12} +$ $95\!\cdots\!24$ $p^{15} T^{13} +$ $38\!\cdots\!08$ $p^{20} T^{14} +$ $90\!\cdots\!72$ $p^{25} T^{15} +$ $38\!\cdots\!93$ $p^{30} T^{16} + 460040940498284 p^{35} T^{17} + 27398194046 p^{40} T^{18} + 1580 p^{45} T^{19} + p^{50} T^{20} )^{2}$
	97	$( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} +$ $98\!\cdots\!25$ $T^{4} -$ $59\!\cdots\!28$ $T^{5} +$ $15\!\cdots\!52$ $T^{6} -$ $87\!\cdots\!16$ $T^{7} +$ $18\!\cdots\!70$ $T^{8} -$ $96\!\cdots\!88$ $T^{9} +$ $17\!\cdots\!72$ $T^{10} -$ $96\!\cdots\!88$ $p^{5} T^{11} +$ $18\!\cdots\!70$ $p^{10} T^{12} -$ $87\!\cdots\!16$ $p^{15} T^{13} +$ $15\!\cdots\!52$ $p^{20} T^{14} -$ $59\!\cdots\!28$ $p^{25} T^{15} +$ $98\!\cdots\!25$ $p^{30} T^{16} - 2817316775448856 p^{35} T^{17} + 43672916862 p^{40} T^{18} - 73688 p^{45} T^{19} + p^{50} T^{20} )^{2}$
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L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−2.01181974853210694069051397527, −2.00184520601613364558815717172, −1.99137701497580300320493208889, −1.86067294221119554808503147766, −1.83528829292558556196671846376, −1.82916402577391776993472198911, −1.71856443665869784807341483005, −1.52327179962858625597132094154, −1.48833237632334566247811975479, −1.47073112052452662358187631910, −1.41388260377538315760294413664, −1.38772710423946496734483801737, −1.35836837350507643172736036392, −1.25858451898299969435584111929, −1.10797962504319227587912313179, −0.949451348844032444099508165619, −0.812333429012405914020304403537, −0.76310990282658015833754615043, −0.68586934983607961592992328049, −0.61052444913038110205597296122, −0.39726206232649474017212139552, −0.33693893195854983889903516442, −0.31206853238131934064917383789, −0.26830791471723154911605044202, −0.05832455248489167330481723962, 0.05832455248489167330481723962, 0.26830791471723154911605044202, 0.31206853238131934064917383789, 0.33693893195854983889903516442, 0.39726206232649474017212139552, 0.61052444913038110205597296122, 0.68586934983607961592992328049, 0.76310990282658015833754615043, 0.812333429012405914020304403537, 0.949451348844032444099508165619, 1.10797962504319227587912313179, 1.25858451898299969435584111929, 1.35836837350507643172736036392, 1.38772710423946496734483801737, 1.41388260377538315760294413664, 1.47073112052452662358187631910, 1.48833237632334566247811975479, 1.52327179962858625597132094154, 1.71856443665869784807341483005, 1.82916402577391776993472198911, 1.83528829292558556196671846376, 1.86067294221119554808503147766, 1.99137701497580300320493208889, 2.00184520601613364558815717172, 2.01181974853210694069051397527

Graph of the $Z$ -function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Imaginary part of the first few zeros on the critical line

Graph of the $Z$ -function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	40-160e20-1.1-c5e20-0-0
Degree	$40$
Conductor	$1.209\times 10^{44}$
Sign	$1$
Analytic cond.	$1.53324\times 10^{28}$
Root an. cond.	$5.06570$
Motivic weight	$5$
Arithmetic	yes
Rational	yes
Primitive	no
Self-dual	yes
Analytic rank	$0$

Properties

Origins

Origins of factors

Downloads

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Particular Values

Imaginary part of the first few zeros on the critical line

Graph of the ZZZ-function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Graph of the $Z$ -function along the critical line