LMFDB - L-function 16-63e8-1.1-c7e8-0-1

Dirichlet series

L(s) = 1

− 2^-s + 206·4^-s + 196·5^-s + 154·7^-s + 607·8^-s − 196·10^-s − 5.21e3·11^-s − 7.58e3·13^-s − 154·14^-s + 2.89e4·16^-s − 1.14e4·17^-s + 6.91e4·19^-s + 4.03e4·20^-s + 5.21e3·22^-s − 1.46e5·23^-s + 1.72e5·25^-s + 7.58e3·26^-s + 3.17e4·28^-s + 1.41e5·29^-s + 2.88e5·31^-s + 1.72e5·32^-s + 1.14e4·34^-s + 3.01e4·35^-s + 4.48e5·37^-s − 6.91e4·38^-s + 1.18e5·40^-s + 1.32e6·41^-s + ⋯

L(s) = 1

− 0.0883·2^-s + 1.60·4^-s + 0.701·5^-s + 0.169·7^-s + 0.419·8^-s − 0.0619·10^-s − 1.18·11^-s − 0.957·13^-s − 0.0149·14^-s + 1.76·16^-s − 0.564·17^-s + 2.31·19^-s + 1.12·20^-s + 0.104·22^-s − 2.50·23^-s + 2.20·25^-s + 0.0846·26^-s + 0.273·28^-s + 1.07·29^-s + 1.74·31^-s + 0.932·32^-s + 0.0498·34^-s + 0.118·35^-s + 1.45·37^-s − 0.204·38^-s + 0.293·40^-s + 3.00·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$3^{16} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$2.25032\times 10^{10}$
Root analytic conductor:	$4.43624$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$62.68054099$
$L(\frac12)$	$\approx$	$62.68054099$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 - 22 p T - 53948 p^{2} T^{2} + 44152 p^{4} T^{3} + 25832189 p^{6} T^{4} + 44152 p^{11} T^{5} - 53948 p^{16} T^{6} - 22 p^{22} T^{7} + p^{28} T^{8} $
good	2	$ 1 + T - 205 T^{2} - 509 p T^{3} + 1463 p^{3} T^{4} + 18021 p^{3} T^{5} + 26303 p^{5} T^{6} - 261523 p^{5} T^{7} - 1566247 p^{6} T^{8} - 261523 p^{12} T^{9} + 26303 p^{19} T^{10} + 18021 p^{24} T^{11} + 1463 p^{31} T^{12} - 509 p^{36} T^{13} - 205 p^{42} T^{14} + p^{49} T^{15} + p^{56} T^{16} $
	5	$ 1 - 196 T - 133927 T^{2} + 582836 p^{3} T^{3} + 176610889 p^{2} T^{4} - 7538169316992 T^{5} + 1573062976649806 T^{6} + 69013603051990576 p T^{7} - 7354408837532411194 p^{2} T^{8} + 69013603051990576 p^{8} T^{9} + 1573062976649806 p^{14} T^{10} - 7538169316992 p^{21} T^{11} + 176610889 p^{30} T^{12} + 582836 p^{38} T^{13} - 133927 p^{42} T^{14} - 196 p^{49} T^{15} + p^{56} T^{16} $
	11	$ 1 + 5210 T - 13550341 T^{2} + 87815127118 T^{3} + 813472841934301 T^{4} - 1670782125020401848 T^{5} + $$73\!\cdots\!98$$ T^{6} + $$52\!\cdots\!28$$ T^{7} - $$17\!\cdots\!38$$ T^{8} + $$52\!\cdots\!28$$ p^{7} T^{9} + $$73\!\cdots\!98$$ p^{14} T^{10} - 1670782125020401848 p^{21} T^{11} + 813472841934301 p^{28} T^{12} + 87815127118 p^{35} T^{13} - 13550341 p^{42} T^{14} + 5210 p^{49} T^{15} + p^{56} T^{16} $
	13	$ ( 1 + 3794 T + 102976741 T^{2} + 172660567546 T^{3} + 8327318022182528 T^{4} + 172660567546 p^{7} T^{5} + 102976741 p^{14} T^{6} + 3794 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	17	$ 1 + 11436 T - 1114853444 T^{2} - 158928045720 T^{3} + 809471792809409914 T^{4} - $$31\!\cdots\!92$$ T^{5} - $$38\!\cdots\!60$$ T^{6} + $$57\!\cdots\!52$$ T^{7} + $$14\!\cdots\!59$$ T^{8} + $$57\!\cdots\!52$$ p^{7} T^{9} - $$38\!\cdots\!60$$ p^{14} T^{10} - $$31\!\cdots\!92$$ p^{21} T^{11} + 809471792809409914 p^{28} T^{12} - 158928045720 p^{35} T^{13} - 1114853444 p^{42} T^{14} + 11436 p^{49} T^{15} + p^{56} T^{16} $
	19	$ 1 - 69158 T + 1595596935 T^{2} - 18293280666090 T^{3} - 81905563961069391 T^{4} + $$37\!\cdots\!96$$ T^{5} - $$13\!\cdots\!74$$ T^{6} + $$11\!\cdots\!36$$ p T^{7} - $$48\!\cdots\!98$$ T^{8} + $$11\!\cdots\!36$$ p^{8} T^{9} - $$13\!\cdots\!74$$ p^{14} T^{10} + $$37\!\cdots\!96$$ p^{21} T^{11} - 81905563961069391 p^{28} T^{12} - 18293280666090 p^{35} T^{13} + 1595596935 p^{42} T^{14} - 69158 p^{49} T^{15} + p^{56} T^{16} $
	23	$ 1 + 146220 T + 1817735620 T^{2} - 252982184818344 T^{3} + 44908344735632175994 T^{4} + $$35\!\cdots\!04$$ T^{5} - $$53\!\cdots\!36$$ T^{6} + $$30\!\cdots\!72$$ T^{7} + $$10\!\cdots\!27$$ T^{8} + $$30\!\cdots\!72$$ p^{7} T^{9} - $$53\!\cdots\!36$$ p^{14} T^{10} + $$35\!\cdots\!04$$ p^{21} T^{11} + 44908344735632175994 p^{28} T^{12} - 252982184818344 p^{35} T^{13} + 1817735620 p^{42} T^{14} + 146220 p^{49} T^{15} + p^{56} T^{16} $
	29	$ ( 1 - 70664 T + 48199790963 T^{2} - 1247069390276928 T^{3} + $$10\!\cdots\!80$$ T^{4} - 1247069390276928 p^{7} T^{5} + 48199790963 p^{14} T^{6} - 70664 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	31	$ 1 - 288618 T - 50722994584 T^{2} + 9961277652025592 T^{3} + $$50\!\cdots\!55$$ T^{4} - $$54\!\cdots\!32$$ T^{5} - $$18\!\cdots\!84$$ T^{6} + $$15\!\cdots\!06$$ T^{7} + $$73\!\cdots\!64$$ T^{8} + $$15\!\cdots\!06$$ p^{7} T^{9} - $$18\!\cdots\!84$$ p^{14} T^{10} - $$54\!\cdots\!32$$ p^{21} T^{11} + $$50\!\cdots\!55$$ p^{28} T^{12} + 9961277652025592 p^{35} T^{13} - 50722994584 p^{42} T^{14} - 288618 p^{49} T^{15} + p^{56} T^{16} $
	37	$ 1 - 448902 T - 84053579929 T^{2} + 1683019295379758 T^{3} + $$60\!\cdots\!09$$ p T^{4} + $$20\!\cdots\!36$$ T^{5} - $$19\!\cdots\!30$$ T^{6} + $$44\!\cdots\!04$$ T^{7} + $$18\!\cdots\!26$$ T^{8} + $$44\!\cdots\!04$$ p^{7} T^{9} - $$19\!\cdots\!30$$ p^{14} T^{10} + $$20\!\cdots\!36$$ p^{21} T^{11} + $$60\!\cdots\!09$$ p^{29} T^{12} + 1683019295379758 p^{35} T^{13} - 84053579929 p^{42} T^{14} - 448902 p^{49} T^{15} + p^{56} T^{16} $
	41	$ ( 1 - 663316 T + 871076913224 T^{2} - 384262293935086044 T^{3} + $$26\!\cdots\!26$$ T^{4} - 384262293935086044 p^{7} T^{5} + 871076913224 p^{14} T^{6} - 663316 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	43	$ ( 1 - 554 T - 15077285 p T^{2} + 4664521444356754 T^{3} + $$13\!\cdots\!28$$ T^{4} + 4664521444356754 p^{7} T^{5} - 15077285 p^{15} T^{6} - 554 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	47	$ 1 - 762180 T - 81239846636 T^{2} - 48896398946940456 T^{3} - $$68\!\cdots\!42$$ T^{4} + $$24\!\cdots\!92$$ T^{5} + $$83\!\cdots\!24$$ T^{6} - $$15\!\cdots\!84$$ T^{7} + $$57\!\cdots\!27$$ T^{8} - $$15\!\cdots\!84$$ p^{7} T^{9} + $$83\!\cdots\!24$$ p^{14} T^{10} + $$24\!\cdots\!92$$ p^{21} T^{11} - $$68\!\cdots\!42$$ p^{28} T^{12} - 48896398946940456 p^{35} T^{13} - 81239846636 p^{42} T^{14} - 762180 p^{49} T^{15} + p^{56} T^{16} $
	53	$ 1 + 2761920 T + 1375852346317 T^{2} - 146941467225378768 T^{3} + $$39\!\cdots\!29$$ T^{4} + $$32\!\cdots\!80$$ T^{5} - $$47\!\cdots\!50$$ T^{6} - $$27\!\cdots\!68$$ T^{7} + $$36\!\cdots\!46$$ T^{8} - $$27\!\cdots\!68$$ p^{7} T^{9} - $$47\!\cdots\!50$$ p^{14} T^{10} + $$32\!\cdots\!80$$ p^{21} T^{11} + $$39\!\cdots\!29$$ p^{28} T^{12} - 146941467225378768 p^{35} T^{13} + 1375852346317 p^{42} T^{14} + 2761920 p^{49} T^{15} + p^{56} T^{16} $
	59	$ 1 - 3410898 T + 868027314559 T^{2} + 5620025039380348794 T^{3} + $$49\!\cdots\!21$$ T^{4} - $$16\!\cdots\!60$$ T^{5} - $$14\!\cdots\!90$$ T^{6} + $$35\!\cdots\!12$$ T^{7} - $$15\!\cdots\!86$$ T^{8} + $$35\!\cdots\!12$$ p^{7} T^{9} - $$14\!\cdots\!90$$ p^{14} T^{10} - $$16\!\cdots\!60$$ p^{21} T^{11} + $$49\!\cdots\!21$$ p^{28} T^{12} + 5620025039380348794 p^{35} T^{13} + 868027314559 p^{42} T^{14} - 3410898 p^{49} T^{15} + p^{56} T^{16} $
	61	$ 1 + 300892 T - 86494008900 p T^{2} + 5853008758722988296 T^{3} + $$12\!\cdots\!58$$ T^{4} - $$28\!\cdots\!44$$ T^{5} + $$20\!\cdots\!08$$ T^{6} + $$60\!\cdots\!48$$ T^{7} - $$94\!\cdots\!09$$ T^{8} + $$60\!\cdots\!48$$ p^{7} T^{9} + $$20\!\cdots\!08$$ p^{14} T^{10} - $$28\!\cdots\!44$$ p^{21} T^{11} + $$12\!\cdots\!58$$ p^{28} T^{12} + 5853008758722988296 p^{35} T^{13} - 86494008900 p^{43} T^{14} + 300892 p^{49} T^{15} + p^{56} T^{16} $
	67	$ 1 - 4222478 T - 9315307834005 T^{2} + 34395757506970837806 T^{3} + $$16\!\cdots\!33$$ T^{4} - $$33\!\cdots\!92$$ T^{5} - $$12\!\cdots\!42$$ T^{6} + $$46\!\cdots\!28$$ T^{7} + $$10\!\cdots\!14$$ T^{8} + $$46\!\cdots\!28$$ p^{7} T^{9} - $$12\!\cdots\!42$$ p^{14} T^{10} - $$33\!\cdots\!92$$ p^{21} T^{11} + $$16\!\cdots\!33$$ p^{28} T^{12} + 34395757506970837806 p^{35} T^{13} - 9315307834005 p^{42} T^{14} - 4222478 p^{49} T^{15} + p^{56} T^{16} $
	71	$ ( 1 - 380964 T + 17482423832204 T^{2} + 29051119312520145372 T^{3} + $$13\!\cdots\!74$$ T^{4} + 29051119312520145372 p^{7} T^{5} + 17482423832204 p^{14} T^{6} - 380964 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	73	$ 1 - 451674 T - 6076887842341 T^{2} - 71774769404461846774 T^{3} - $$23\!\cdots\!35$$ T^{4} + $$66\!\cdots\!80$$ T^{5} + $$24\!\cdots\!22$$ T^{6} - $$30\!\cdots\!80$$ T^{7} - $$26\!\cdots\!38$$ T^{8} - $$30\!\cdots\!80$$ p^{7} T^{9} + $$24\!\cdots\!22$$ p^{14} T^{10} + $$66\!\cdots\!80$$ p^{21} T^{11} - $$23\!\cdots\!35$$ p^{28} T^{12} - 71774769404461846774 p^{35} T^{13} - 6076887842341 p^{42} T^{14} - 451674 p^{49} T^{15} + p^{56} T^{16} $
	79	$ 1 - 12154822 T + 90170014725672 T^{2} - $$60\!\cdots\!68$$ T^{3} + $$31\!\cdots\!03$$ T^{4} - $$13\!\cdots\!68$$ T^{5} + $$58\!\cdots\!24$$ T^{6} - $$22\!\cdots\!34$$ T^{7} + $$87\!\cdots\!36$$ T^{8} - $$22\!\cdots\!34$$ p^{7} T^{9} + $$58\!\cdots\!24$$ p^{14} T^{10} - $$13\!\cdots\!68$$ p^{21} T^{11} + $$31\!\cdots\!03$$ p^{28} T^{12} - $$60\!\cdots\!68$$ p^{35} T^{13} + 90170014725672 p^{42} T^{14} - 12154822 p^{49} T^{15} + p^{56} T^{16} $
	83	$ ( 1 - 12087978 T + 133280044237529 T^{2} - $$88\!\cdots\!42$$ T^{3} + $$55\!\cdots\!20$$ T^{4} - $$88\!\cdots\!42$$ p^{7} T^{5} + 133280044237529 p^{14} T^{6} - 12087978 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	89	$ 1 - 4955752 T - 151120762805968 T^{2} + $$40\!\cdots\!16$$ T^{3} + $$16\!\cdots\!34$$ T^{4} - $$25\!\cdots\!20$$ T^{5} - $$10\!\cdots\!16$$ T^{6} + $$37\!\cdots\!48$$ T^{7} + $$56\!\cdots\!87$$ T^{8} + $$37\!\cdots\!48$$ p^{7} T^{9} - $$10\!\cdots\!16$$ p^{14} T^{10} - $$25\!\cdots\!20$$ p^{21} T^{11} + $$16\!\cdots\!34$$ p^{28} T^{12} + $$40\!\cdots\!16$$ p^{35} T^{13} - 151120762805968 p^{42} T^{14} - 4955752 p^{49} T^{15} + p^{56} T^{16} $
	97	$ ( 1 + 22840614 T + 516597355973305 T^{2} + $$62\!\cdots\!78$$ T^{3} + $$71\!\cdots\!48$$ T^{4} + $$62\!\cdots\!78$$ p^{7} T^{5} + 516597355973305 p^{14} T^{6} + 22840614 p^{21} T^{7} + p^{28} T^{8} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.51042673816867756724809834414, −5.33910294454682421113255651658, −5.12364182304182091875626329614, −5.05306169097531726510222605106, −4.67664871950749303115676327737, −4.61714282873216234169376470025, −4.48779838242372634312711509977, −4.11723385155902819661124047961, −3.88249561024902311952116057478, −3.71436063776737489672777046960, −3.39054700781508067773144333223, −3.32015849081514474601455111043, −2.77558979408361996243685593576, −2.59319370732128406579054309435, −2.57269045227894597999459053235, −2.44242764082802598078629851268, −2.41977105557665952518090115422, −1.91766581453537347328269316365, −1.86369500206933907292338679609, −1.40630528192828701414920454954, −1.13000944792508839656636016535, −0.823450308388871391058056931353, −0.64934329427361870445977549321, −0.64860142078860972552518359003, −0.51056083655295912217857597277, 0.51056083655295912217857597277, 0.64860142078860972552518359003, 0.64934329427361870445977549321, 0.823450308388871391058056931353, 1.13000944792508839656636016535, 1.40630528192828701414920454954, 1.86369500206933907292338679609, 1.91766581453537347328269316365, 2.41977105557665952518090115422, 2.44242764082802598078629851268, 2.57269045227894597999459053235, 2.59319370732128406579054309435, 2.77558979408361996243685593576, 3.32015849081514474601455111043, 3.39054700781508067773144333223, 3.71436063776737489672777046960, 3.88249561024902311952116057478, 4.11723385155902819661124047961, 4.48779838242372634312711509977, 4.61714282873216234169376470025, 4.67664871950749303115676327737, 5.05306169097531726510222605106, 5.12364182304182091875626329614, 5.33910294454682421113255651658, 5.51042673816867756724809834414

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(62.68054099\)
\(L(\frac12)\)	\(\approx\)	\(62.68054099\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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