LMFDB - L-function 18-325e9-1.1-c5e9-0-0

Dirichlet series

L(s) = 1

+ 5·2^-s + 11·3^-s − 86·4^-s + 55·6^-s − 12·7^-s − 312·8^-s − 752·9^-s − 1.42e3·11^-s − 946·12^-s − 1.52e3·13^-s − 60·14^-s + 3.45e3·16^-s − 648·17^-s − 3.76e3·18^-s − 408·19^-s − 132·21^-s − 7.11e3·22^-s − 1.83e3·23^-s − 3.43e3·24^-s − 7.60e3·26^-s − 7.05e3·27^-s + 1.03e3·28^-s − 8.73e3·29^-s + 748·31^-s − 473·32^-s − 1.56e4·33^-s − 3.24e3·34^-s + ⋯

L(s) = 1

+ 0.883·2^-s + 0.705·3^-s − 2.68·4^-s + 0.623·6^-s − 0.0925·7^-s − 1.72·8^-s − 3.09·9^-s − 3.54·11^-s − 1.89·12^-s − 2.49·13^-s − 0.0818·14^-s + 3.37·16^-s − 0.543·17^-s − 2.73·18^-s − 0.259·19^-s − 0.0653·21^-s − 3.13·22^-s − 0.724·23^-s − 1.21·24^-s − 2.20·26^-s − 1.86·27^-s + 0.248·28^-s − 1.92·29^-s + 0.139·31^-s − 0.0816·32^-s − 2.50·33^-s − 0.480·34^-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree:	$18$
Conductor:	$5^{18} \cdot 13^{9}$
Sign:	$-1$
Analytic conductor:	$2.84050\times 10^{15}$
Root analytic conductor:	$7.21974$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$9$
Selberg data:	$(18,\ 5^{18} \cdot 13^{9} ,\ ( \ : [5/2]^{9} ),\ -1 )$

Particular Values

$L(3)$	$=$	$0$
$L(\frac12)$	$=$	$0$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	13	$( 1 + p^{2} T )^{9}$
good	2	$1 - 5 T + 111 T^{2} - 673 T^{3} + 1975 p^{2} T^{4} - 22509 p T^{5} + 104647 p^{2} T^{6} - 271961 p^{3} T^{7} + 1046245 p^{4} T^{8} - 633137 p^{7} T^{9} + 1046245 p^{9} T^{10} - 271961 p^{13} T^{11} + 104647 p^{17} T^{12} - 22509 p^{21} T^{13} + 1975 p^{27} T^{14} - 673 p^{30} T^{15} + 111 p^{35} T^{16} - 5 p^{40} T^{17} + p^{45} T^{18}$
	3	$1 - 11 T + 97 p^{2} T^{2} - 1202 p^{2} T^{3} + 149905 p T^{4} - 2119657 p T^{5} + 162917153 T^{6} - 2502982570 T^{7} + 15828224858 p T^{8} - 79712893852 p^{2} T^{9} + 15828224858 p^{6} T^{10} - 2502982570 p^{10} T^{11} + 162917153 p^{15} T^{12} - 2119657 p^{21} T^{13} + 149905 p^{26} T^{14} - 1202 p^{32} T^{15} + 97 p^{37} T^{16} - 11 p^{40} T^{17} + p^{45} T^{18}$
	7	$1 + 12 T + 96157 T^{2} + 562396 T^{3} + 4524975747 T^{4} - 9279133840 T^{5} + 138615511394837 T^{6} - 1012284041916540 T^{7} + 3074128450205785774 T^{8} - 25354172273837803832 T^{9} + 3074128450205785774 p^{5} T^{10} - 1012284041916540 p^{10} T^{11} + 138615511394837 p^{15} T^{12} - 9279133840 p^{20} T^{13} + 4524975747 p^{25} T^{14} + 562396 p^{30} T^{15} + 96157 p^{35} T^{16} + 12 p^{40} T^{17} + p^{45} T^{18}$
	11	$1 + 1422 T + 1678928 T^{2} + 1457218782 T^{3} + 1113094396148 T^{4} + 714749702584070 T^{5} + 415158952053473389 T^{6} +$ $21\!\cdots\!24$ $T^{7} +$ $90\!\cdots\!88$ $p T^{8} +$ $34\!\cdots\!96$ $p^{2} T^{9} +$ $90\!\cdots\!88$ $p^{6} T^{10} +$ $21\!\cdots\!24$ $p^{10} T^{11} + 415158952053473389 p^{15} T^{12} + 714749702584070 p^{20} T^{13} + 1113094396148 p^{25} T^{14} + 1457218782 p^{30} T^{15} + 1678928 p^{35} T^{16} + 1422 p^{40} T^{17} + p^{45} T^{18}$
	17	$1 + 648 T + 3847624 T^{2} + 3743716930 T^{3} + 7955206965208 T^{4} + 9997321106254216 T^{5} + 16234048035213331295 T^{6} +$ $17\!\cdots\!84$ $T^{7} +$ $31\!\cdots\!88$ $T^{8} +$ $26\!\cdots\!20$ $T^{9} +$ $31\!\cdots\!88$ $p^{5} T^{10} +$ $17\!\cdots\!84$ $p^{10} T^{11} + 16234048035213331295 p^{15} T^{12} + 9997321106254216 p^{20} T^{13} + 7955206965208 p^{25} T^{14} + 3743716930 p^{30} T^{15} + 3847624 p^{35} T^{16} + 648 p^{40} T^{17} + p^{45} T^{18}$
	19	$1 + 408 T + 13870454 T^{2} + 362063268 p T^{3} + 96344427116891 T^{4} + 49973344519545016 T^{5} +$ $44\!\cdots\!51$ $T^{6} +$ $21\!\cdots\!32$ $T^{7} +$ $14\!\cdots\!93$ $T^{8} +$ $33\!\cdots\!80$ $p T^{9} +$ $14\!\cdots\!93$ $p^{5} T^{10} +$ $21\!\cdots\!32$ $p^{10} T^{11} +$ $44\!\cdots\!51$ $p^{15} T^{12} + 49973344519545016 p^{20} T^{13} + 96344427116891 p^{25} T^{14} + 362063268 p^{31} T^{15} + 13870454 p^{35} T^{16} + 408 p^{40} T^{17} + p^{45} T^{18}$
	23	$1 + 1839 T + 27570110 T^{2} + 54947371329 T^{3} + 393408475518083 T^{4} + 689356483695756566 T^{5} +$ $38\!\cdots\!67$ $T^{6} +$ $56\!\cdots\!07$ $T^{7} +$ $28\!\cdots\!41$ $T^{8} +$ $38\!\cdots\!86$ $T^{9} +$ $28\!\cdots\!41$ $p^{5} T^{10} +$ $56\!\cdots\!07$ $p^{10} T^{11} +$ $38\!\cdots\!67$ $p^{15} T^{12} + 689356483695756566 p^{20} T^{13} + 393408475518083 p^{25} T^{14} + 54947371329 p^{30} T^{15} + 27570110 p^{35} T^{16} + 1839 p^{40} T^{17} + p^{45} T^{18}$
	29	$1 + 8737 T + 100263490 T^{2} + 812860482637 T^{3} + 5877575658032654 T^{4} + 37024407518884704661 T^{5} +$ $22\!\cdots\!59$ $T^{6} +$ $11\!\cdots\!90$ $T^{7} +$ $61\!\cdots\!04$ $T^{8} +$ $28\!\cdots\!94$ $T^{9} +$ $61\!\cdots\!04$ $p^{5} T^{10} +$ $11\!\cdots\!90$ $p^{10} T^{11} +$ $22\!\cdots\!59$ $p^{15} T^{12} + 37024407518884704661 p^{20} T^{13} + 5877575658032654 p^{25} T^{14} + 812860482637 p^{30} T^{15} + 100263490 p^{35} T^{16} + 8737 p^{40} T^{17} + p^{45} T^{18}$
	31	$1 - 748 T + 75897937 T^{2} + 87335748108 T^{3} + 3911511988579655 T^{4} + 5672812818384581976 T^{5} +$ $17\!\cdots\!49$ $T^{6} +$ $26\!\cdots\!84$ $T^{7} +$ $56\!\cdots\!86$ $T^{8} +$ $10\!\cdots\!68$ $T^{9} +$ $56\!\cdots\!86$ $p^{5} T^{10} +$ $26\!\cdots\!84$ $p^{10} T^{11} +$ $17\!\cdots\!49$ $p^{15} T^{12} + 5672812818384581976 p^{20} T^{13} + 3911511988579655 p^{25} T^{14} + 87335748108 p^{30} T^{15} + 75897937 p^{35} T^{16} - 748 p^{40} T^{17} + p^{45} T^{18}$
	37	$1 - 15486 T + 406034081 T^{2} - 4455267036680 T^{3} + 69627769486559328 T^{4} -$ $58\!\cdots\!48$ $T^{5} +$ $71\!\cdots\!32$ $T^{6} -$ $48\!\cdots\!40$ $T^{7} +$ $53\!\cdots\!30$ $T^{8} -$ $33\!\cdots\!88$ $T^{9} +$ $53\!\cdots\!30$ $p^{5} T^{10} -$ $48\!\cdots\!40$ $p^{10} T^{11} +$ $71\!\cdots\!32$ $p^{15} T^{12} -$ $58\!\cdots\!48$ $p^{20} T^{13} + 69627769486559328 p^{25} T^{14} - 4455267036680 p^{30} T^{15} + 406034081 p^{35} T^{16} - 15486 p^{40} T^{17} + p^{45} T^{18}$
	41	$1 + 28676 T + 807443290 T^{2} + 15162634916010 T^{3} + 279540162665697191 T^{4} +$ $41\!\cdots\!68$ $T^{5} +$ $61\!\cdots\!69$ $T^{6} +$ $77\!\cdots\!70$ $T^{7} +$ $96\!\cdots\!69$ $T^{8} +$ $10\!\cdots\!84$ $T^{9} +$ $96\!\cdots\!69$ $p^{5} T^{10} +$ $77\!\cdots\!70$ $p^{10} T^{11} +$ $61\!\cdots\!69$ $p^{15} T^{12} +$ $41\!\cdots\!68$ $p^{20} T^{13} + 279540162665697191 p^{25} T^{14} + 15162634916010 p^{30} T^{15} + 807443290 p^{35} T^{16} + 28676 p^{40} T^{17} + p^{45} T^{18}$
	43	$1 + 28665 T + 905859098 T^{2} + 19568343598455 T^{3} + 410275978472684999 T^{4} +$ $70\!\cdots\!58$ $T^{5} +$ $11\!\cdots\!11$ $T^{6} +$ $16\!\cdots\!25$ $T^{7} +$ $23\!\cdots\!49$ $T^{8} +$ $28\!\cdots\!26$ $T^{9} +$ $23\!\cdots\!49$ $p^{5} T^{10} +$ $16\!\cdots\!25$ $p^{10} T^{11} +$ $11\!\cdots\!11$ $p^{15} T^{12} +$ $70\!\cdots\!58$ $p^{20} T^{13} + 410275978472684999 p^{25} T^{14} + 19568343598455 p^{30} T^{15} + 905859098 p^{35} T^{16} + 28665 p^{40} T^{17} + p^{45} T^{18}$
	47	$1 - 29452 T + 1346699157 T^{2} - 23094731728324 T^{3} + 695077981104733875 T^{4} -$ $92\!\cdots\!92$ $T^{5} +$ $25\!\cdots\!29$ $T^{6} -$ $31\!\cdots\!36$ $T^{7} +$ $77\!\cdots\!42$ $T^{8} -$ $82\!\cdots\!48$ $T^{9} +$ $77\!\cdots\!42$ $p^{5} T^{10} -$ $31\!\cdots\!36$ $p^{10} T^{11} +$ $25\!\cdots\!29$ $p^{15} T^{12} -$ $92\!\cdots\!92$ $p^{20} T^{13} + 695077981104733875 p^{25} T^{14} - 23094731728324 p^{30} T^{15} + 1346699157 p^{35} T^{16} - 29452 p^{40} T^{17} + p^{45} T^{18}$
	53	$1 + 75977 T + 4311797066 T^{2} + 190721914335121 T^{3} + 7007646765751779502 T^{4} +$ $22\!\cdots\!57$ $T^{5} +$ $64\!\cdots\!47$ $T^{6} +$ $16\!\cdots\!26$ $T^{7} +$ $38\!\cdots\!24$ $T^{8} +$ $82\!\cdots\!74$ $T^{9} +$ $38\!\cdots\!24$ $p^{5} T^{10} +$ $16\!\cdots\!26$ $p^{10} T^{11} +$ $64\!\cdots\!47$ $p^{15} T^{12} +$ $22\!\cdots\!57$ $p^{20} T^{13} + 7007646765751779502 p^{25} T^{14} + 190721914335121 p^{30} T^{15} + 4311797066 p^{35} T^{16} + 75977 p^{40} T^{17} + p^{45} T^{18}$
	59	$1 + 88142 T + 7408012125 T^{2} + 418488720267288 T^{3} + 21514792833002089743 T^{4} +$ $91\!\cdots\!78$ $T^{5} +$ $35\!\cdots\!37$ $T^{6} +$ $11\!\cdots\!40$ $T^{7} +$ $37\!\cdots\!18$ $T^{8} +$ $10\!\cdots\!16$ $T^{9} +$ $37\!\cdots\!18$ $p^{5} T^{10} +$ $11\!\cdots\!40$ $p^{10} T^{11} +$ $35\!\cdots\!37$ $p^{15} T^{12} +$ $91\!\cdots\!78$ $p^{20} T^{13} + 21514792833002089743 p^{25} T^{14} + 418488720267288 p^{30} T^{15} + 7408012125 p^{35} T^{16} + 88142 p^{40} T^{17} + p^{45} T^{18}$
	61	$1 - 28165 T + 5525406634 T^{2} - 111231076019257 T^{3} + 13564817030465760102 T^{4} -$ $20\!\cdots\!21$ $T^{5} +$ $20\!\cdots\!83$ $T^{6} -$ $23\!\cdots\!82$ $T^{7} +$ $23\!\cdots\!32$ $T^{8} -$ $22\!\cdots\!54$ $T^{9} +$ $23\!\cdots\!32$ $p^{5} T^{10} -$ $23\!\cdots\!82$ $p^{10} T^{11} +$ $20\!\cdots\!83$ $p^{15} T^{12} -$ $20\!\cdots\!21$ $p^{20} T^{13} + 13564817030465760102 p^{25} T^{14} - 111231076019257 p^{30} T^{15} + 5525406634 p^{35} T^{16} - 28165 p^{40} T^{17} + p^{45} T^{18}$
	67	$1 - 94754 T + 14705440952 T^{2} - 1016307598907474 T^{3} + 88638330878987469708 T^{4} -$ $47\!\cdots\!90$ $T^{5} +$ $29\!\cdots\!49$ $T^{6} -$ $12\!\cdots\!96$ $T^{7} +$ $61\!\cdots\!56$ $T^{8} -$ $21\!\cdots\!48$ $T^{9} +$ $61\!\cdots\!56$ $p^{5} T^{10} -$ $12\!\cdots\!96$ $p^{10} T^{11} +$ $29\!\cdots\!49$ $p^{15} T^{12} -$ $47\!\cdots\!90$ $p^{20} T^{13} + 88638330878987469708 p^{25} T^{14} - 1016307598907474 p^{30} T^{15} + 14705440952 p^{35} T^{16} - 94754 p^{40} T^{17} + p^{45} T^{18}$
	71	$1 + 70562 T + 145680381 p T^{2} + 535896116798952 T^{3} + 45814913528036188080 T^{4} +$ $19\!\cdots\!72$ $T^{5} +$ $12\!\cdots\!20$ $T^{6} +$ $50\!\cdots\!60$ $T^{7} +$ $27\!\cdots\!26$ $T^{8} +$ $98\!\cdots\!80$ $T^{9} +$ $27\!\cdots\!26$ $p^{5} T^{10} +$ $50\!\cdots\!60$ $p^{10} T^{11} +$ $12\!\cdots\!20$ $p^{15} T^{12} +$ $19\!\cdots\!72$ $p^{20} T^{13} + 45814913528036188080 p^{25} T^{14} + 535896116798952 p^{30} T^{15} + 145680381 p^{36} T^{16} + 70562 p^{40} T^{17} + p^{45} T^{18}$
	73	$1 + 60602 T + 11519959226 T^{2} + 450972296343384 T^{3} + 50569765709477765011 T^{4} +$ $90\!\cdots\!56$ $T^{5} +$ $10\!\cdots\!77$ $T^{6} -$ $13\!\cdots\!72$ $T^{7} +$ $10\!\cdots\!17$ $T^{8} -$ $74\!\cdots\!16$ $T^{9} +$ $10\!\cdots\!17$ $p^{5} T^{10} -$ $13\!\cdots\!72$ $p^{10} T^{11} +$ $10\!\cdots\!77$ $p^{15} T^{12} +$ $90\!\cdots\!56$ $p^{20} T^{13} + 50569765709477765011 p^{25} T^{14} + 450972296343384 p^{30} T^{15} + 11519959226 p^{35} T^{16} + 60602 p^{40} T^{17} + p^{45} T^{18}$
	79	$1 + 164073 T + 28189254670 T^{2} + 2711262477059871 T^{3} +$ $27\!\cdots\!51$ $T^{4} +$ $19\!\cdots\!66$ $T^{5} +$ $15\!\cdots\!63$ $T^{6} +$ $95\!\cdots\!53$ $T^{7} +$ $64\!\cdots\!81$ $T^{8} +$ $34\!\cdots\!70$ $T^{9} +$ $64\!\cdots\!81$ $p^{5} T^{10} +$ $95\!\cdots\!53$ $p^{10} T^{11} +$ $15\!\cdots\!63$ $p^{15} T^{12} +$ $19\!\cdots\!66$ $p^{20} T^{13} +$ $27\!\cdots\!51$ $p^{25} T^{14} + 2711262477059871 p^{30} T^{15} + 28189254670 p^{35} T^{16} + 164073 p^{40} T^{17} + p^{45} T^{18}$
	83	$1 - 22458 T + 4152756288 T^{2} + 65474123064726 T^{3} + 17405654864177049364 T^{4} -$ $56\!\cdots\!58$ $T^{5} +$ $10\!\cdots\!41$ $T^{6} -$ $41\!\cdots\!84$ $T^{7} +$ $20\!\cdots\!72$ $T^{8} -$ $10\!\cdots\!28$ $T^{9} +$ $20\!\cdots\!72$ $p^{5} T^{10} -$ $41\!\cdots\!84$ $p^{10} T^{11} +$ $10\!\cdots\!41$ $p^{15} T^{12} -$ $56\!\cdots\!58$ $p^{20} T^{13} + 17405654864177049364 p^{25} T^{14} + 65474123064726 p^{30} T^{15} + 4152756288 p^{35} T^{16} - 22458 p^{40} T^{17} + p^{45} T^{18}$
	89	$1 + 252698 T + 59335278754 T^{2} + 8555077606937916 T^{3} +$ $11\!\cdots\!11$ $T^{4} +$ $11\!\cdots\!76$ $T^{5} +$ $10\!\cdots\!97$ $T^{6} +$ $77\!\cdots\!76$ $T^{7} +$ $59\!\cdots\!77$ $T^{8} +$ $42\!\cdots\!68$ $T^{9} +$ $59\!\cdots\!77$ $p^{5} T^{10} +$ $77\!\cdots\!76$ $p^{10} T^{11} +$ $10\!\cdots\!97$ $p^{15} T^{12} +$ $11\!\cdots\!76$ $p^{20} T^{13} +$ $11\!\cdots\!11$ $p^{25} T^{14} + 8555077606937916 p^{30} T^{15} + 59335278754 p^{35} T^{16} + 252698 p^{40} T^{17} + p^{45} T^{18}$
	97	$1 + 137986 T + 49335221217 T^{2} + 4614937388745376 T^{3} +$ $90\!\cdots\!56$ $T^{4} +$ $48\!\cdots\!84$ $T^{5} +$ $72\!\cdots\!68$ $T^{6} -$ $13\!\cdots\!72$ $T^{7} +$ $22\!\cdots\!46$ $T^{8} -$ $30\!\cdots\!76$ $T^{9} +$ $22\!\cdots\!46$ $p^{5} T^{10} -$ $13\!\cdots\!72$ $p^{10} T^{11} +$ $72\!\cdots\!68$ $p^{15} T^{12} +$ $48\!\cdots\!84$ $p^{20} T^{13} +$ $90\!\cdots\!56$ $p^{25} T^{14} + 4614937388745376 p^{30} T^{15} + 49335221217 p^{35} T^{16} + 137986 p^{40} T^{17} + p^{45} T^{18}$
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L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.42273691739669564308062997730, −4.37682272114799225031251111598, −4.30510933208574607435478014981, −4.15468565316900567701876607249, −4.04104828521342346299089149679, −3.84427667092487168652050435055, −3.73429801663297766238218452972, −3.26883537432043977553349622581, −3.16393518645599509801474451960, −3.13403914681743963082094512343, −3.11002238949483094823274849681, −3.10717132812192054745819131849, −2.78727354944167646215676148119, −2.76961448391811535394608730749, −2.73814101749413563883652858634, −2.35462796937034092988099424967, −2.27852694118193300828216366092, −2.14971235139531036061675877744, −1.86663097072994345048910443932, −1.79831990409123982812773120552, −1.69344998048524928618759341729, −1.34283860901037459188242718221, −1.16944459963800555915730240496, −1.16903717240384284724983499287, −0.817923808496536573826632166145, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.817923808496536573826632166145, 1.16903717240384284724983499287, 1.16944459963800555915730240496, 1.34283860901037459188242718221, 1.69344998048524928618759341729, 1.79831990409123982812773120552, 1.86663097072994345048910443932, 2.14971235139531036061675877744, 2.27852694118193300828216366092, 2.35462796937034092988099424967, 2.73814101749413563883652858634, 2.76961448391811535394608730749, 2.78727354944167646215676148119, 3.10717132812192054745819131849, 3.11002238949483094823274849681, 3.13403914681743963082094512343, 3.16393518645599509801474451960, 3.26883537432043977553349622581, 3.73429801663297766238218452972, 3.84427667092487168652050435055, 4.04104828521342346299089149679, 4.15468565316900567701876607249, 4.30510933208574607435478014981, 4.37682272114799225031251111598, 4.42273691739669564308062997730

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

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Dirichlet series

Functional equation

Invariants

Particular Values

Euler product

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	18-325e9-1.1-c5e9-0-0
Degree	$18$
Conductor	$4.045\times 10^{22}$
Sign	$-1$
Analytic cond.	$2.84050\times 10^{15}$
Root an. cond.	$7.21974$
Motivic weight	$5$
Arithmetic	yes
Rational	yes
Primitive	no
Self-dual	yes
Analytic rank	$9$

Properties

Origins

Origins of factors

Downloads

Learn more

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