LMFDB - L-function 8-2e4-1.1-c73e4-0-0

Dirichlet series

L(s) = 1

+ 2.74e11·2^-s + 3.05e17·3^-s + 4.72e22·4^-s − 7.84e24·5^-s + 8.38e28·6^-s + 3.63e30·7^-s + 6.49e33·8^-s − 2.74e34·9^-s − 2.15e36·10^-s + 5.27e37·11^-s + 1.44e40·12^-s + 1.35e39·13^-s + 9.99e41·14^-s − 2.39e42·15^-s + 7.80e44·16^-s + 9.54e44·17^-s − 7.54e45·18^-s + 9.54e46·19^-s − 3.70e47·20^-s + 1.10e48·21^-s + 1.45e49·22^-s − 6.74e49·23^-s + 1.97e51·24^-s − 3.56e50·25^-s + 3.72e50·26^-s − 2.06e52·27^-s + 1.71e53·28^-s + ⋯

L(s) = 1

+ 2.82·2^-s + 1.17·3^-s + 5·4^-s − 0.241·5^-s + 3.31·6^-s + 0.518·7^-s + 7.07·8^-s − 0.406·9^-s − 0.682·10^-s + 0.514·11^-s + 5.86·12^-s + 0.0297·13^-s + 1.46·14^-s − 0.282·15^-s + 35/4·16^-s + 1.17·17^-s − 1.14·18^-s + 2.01·19^-s − 1.20·20^-s + 0.608·21^-s + 1.45·22^-s − 1.33·23^-s + 8.29·24^-s − 0.336·25^-s + 0.0840·26^-s − 1.17·27^-s + 2.59·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(74-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+73/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$16$ = $2^{4}$
Sign:	$1$
Analytic conductor:	$2.07554\times 10^{7}$
Root analytic conductor:	$8.21564$
Motivic weight:	$73$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 16,\ (\ :73/2, 73/2, 73/2, 73/2),\ 1)$

Particular Values

$L(37)$	$\approx$	$154.8346583$
$L(\frac12)$	$\approx$	$154.8346583$
$L(\frac{75}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2	$C_1$	$ ( 1 - p^{36} T )^{4} $
good	3	$C_2 \wr S_4$	$ 1 - 101688775241081392 p T + $$61\!\cdots\!76$$ p^{9} T^{2} - $$21\!\cdots\!40$$ p^{19} T^{3} + $$18\!\cdots\!62$$ p^{37} T^{4} - $$21\!\cdots\!40$$ p^{92} T^{5} + $$61\!\cdots\!76$$ p^{155} T^{6} - 101688775241081392 p^{220} T^{7} + p^{292} T^{8} $
	5	$C_2 \wr S_4$	$ 1 + $$15\!\cdots\!92$$ p T + $$13\!\cdots\!76$$ p^{5} T^{2} - $$14\!\cdots\!12$$ p^{13} T^{3} - $$49\!\cdots\!66$$ p^{25} T^{4} - $$14\!\cdots\!12$$ p^{86} T^{5} + $$13\!\cdots\!76$$ p^{151} T^{6} + $$15\!\cdots\!92$$ p^{220} T^{7} + p^{292} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - $$51\!\cdots\!16$$ p T + $$47\!\cdots\!76$$ p^{5} T^{2} + $$10\!\cdots\!80$$ p^{11} T^{3} + $$28\!\cdots\!06$$ p^{20} T^{4} + $$10\!\cdots\!80$$ p^{84} T^{5} + $$47\!\cdots\!76$$ p^{151} T^{6} - $$51\!\cdots\!16$$ p^{220} T^{7} + p^{292} T^{8} $
	11	$C_2 \wr S_4$	$ 1 - $$47\!\cdots\!68$$ p T + $$15\!\cdots\!28$$ p^{2} T^{2} + $$25\!\cdots\!44$$ p^{7} T^{3} + $$40\!\cdots\!70$$ p^{13} T^{4} + $$25\!\cdots\!44$$ p^{80} T^{5} + $$15\!\cdots\!28$$ p^{148} T^{6} - $$47\!\cdots\!68$$ p^{220} T^{7} + p^{292} T^{8} $
	13	$C_2 \wr S_4$	$ 1 - $$13\!\cdots\!76$$ T + $$15\!\cdots\!56$$ p T^{2} - $$66\!\cdots\!20$$ p^{4} T^{3} + $$75\!\cdots\!02$$ p^{9} T^{4} - $$66\!\cdots\!20$$ p^{77} T^{5} + $$15\!\cdots\!56$$ p^{147} T^{6} - $$13\!\cdots\!76$$ p^{219} T^{7} + p^{292} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - $$95\!\cdots\!52$$ T + $$14\!\cdots\!36$$ p T^{2} - $$20\!\cdots\!60$$ p^{4} T^{3} + $$34\!\cdots\!46$$ p^{8} T^{4} - $$20\!\cdots\!60$$ p^{77} T^{5} + $$14\!\cdots\!36$$ p^{147} T^{6} - $$95\!\cdots\!52$$ p^{219} T^{7} + p^{292} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - $$95\!\cdots\!00$$ T + $$54\!\cdots\!44$$ p T^{2} - $$81\!\cdots\!00$$ p^{3} T^{3} + $$73\!\cdots\!06$$ p^{6} T^{4} - $$81\!\cdots\!00$$ p^{76} T^{5} + $$54\!\cdots\!44$$ p^{147} T^{6} - $$95\!\cdots\!00$$ p^{219} T^{7} + p^{292} T^{8} $
	23	$C_2 \wr S_4$	$ 1 + $$29\!\cdots\!48$$ p T + $$33\!\cdots\!64$$ p^{3} T^{2} + $$17\!\cdots\!00$$ p^{6} T^{3} + $$10\!\cdots\!62$$ p^{9} T^{4} + $$17\!\cdots\!00$$ p^{79} T^{5} + $$33\!\cdots\!64$$ p^{149} T^{6} + $$29\!\cdots\!48$$ p^{220} T^{7} + p^{292} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + $$14\!\cdots\!60$$ p T + $$78\!\cdots\!64$$ p T^{2} + $$27\!\cdots\!20$$ p^{3} T^{3} + $$94\!\cdots\!74$$ p^{5} T^{4} + $$27\!\cdots\!20$$ p^{76} T^{5} + $$78\!\cdots\!64$$ p^{147} T^{6} + $$14\!\cdots\!60$$ p^{220} T^{7} + p^{292} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - $$37\!\cdots\!08$$ T + $$57\!\cdots\!48$$ p T^{2} - $$21\!\cdots\!76$$ p^{3} T^{3} + $$54\!\cdots\!70$$ p^{5} T^{4} - $$21\!\cdots\!76$$ p^{76} T^{5} + $$57\!\cdots\!48$$ p^{147} T^{6} - $$37\!\cdots\!08$$ p^{219} T^{7} + p^{292} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - $$68\!\cdots\!52$$ T + $$27\!\cdots\!52$$ T^{2} - $$19\!\cdots\!60$$ p T^{3} + $$10\!\cdots\!74$$ p^{2} T^{4} - $$19\!\cdots\!60$$ p^{74} T^{5} + $$27\!\cdots\!52$$ p^{146} T^{6} - $$68\!\cdots\!52$$ p^{219} T^{7} + p^{292} T^{8} $
	41	$C_2 \wr S_4$	$ 1 - $$56\!\cdots\!48$$ T + $$97\!\cdots\!48$$ T^{2} - $$25\!\cdots\!96$$ p T^{3} + $$34\!\cdots\!70$$ p^{2} T^{4} - $$25\!\cdots\!96$$ p^{74} T^{5} + $$97\!\cdots\!48$$ p^{146} T^{6} - $$56\!\cdots\!48$$ p^{219} T^{7} + p^{292} T^{8} $
	43	$C_2 \wr S_4$	$ 1 + $$19\!\cdots\!04$$ T + $$65\!\cdots\!28$$ T^{2} + $$21\!\cdots\!40$$ p T^{3} + $$90\!\cdots\!14$$ p^{2} T^{4} + $$21\!\cdots\!40$$ p^{74} T^{5} + $$65\!\cdots\!28$$ p^{146} T^{6} + $$19\!\cdots\!04$$ p^{219} T^{7} + p^{292} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + $$26\!\cdots\!88$$ T + $$68\!\cdots\!96$$ p T^{2} + $$76\!\cdots\!80$$ p^{2} T^{3} + $$58\!\cdots\!02$$ p^{3} T^{4} + $$76\!\cdots\!80$$ p^{75} T^{5} + $$68\!\cdots\!96$$ p^{147} T^{6} + $$26\!\cdots\!88$$ p^{219} T^{7} + p^{292} T^{8} $
	53	$C_2 \wr S_4$	$ 1 - $$45\!\cdots\!16$$ T + $$22\!\cdots\!96$$ p T^{2} - $$30\!\cdots\!40$$ p^{2} T^{3} + $$47\!\cdots\!98$$ p^{3} T^{4} - $$30\!\cdots\!40$$ p^{75} T^{5} + $$22\!\cdots\!96$$ p^{147} T^{6} - $$45\!\cdots\!16$$ p^{219} T^{7} + p^{292} T^{8} $
	59	$C_2 \wr S_4$	$ 1 + $$36\!\cdots\!80$$ T + $$81\!\cdots\!24$$ p T^{2} + $$37\!\cdots\!60$$ p^{2} T^{3} + $$56\!\cdots\!74$$ p^{3} T^{4} + $$37\!\cdots\!60$$ p^{75} T^{5} + $$81\!\cdots\!24$$ p^{147} T^{6} + $$36\!\cdots\!80$$ p^{219} T^{7} + p^{292} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - $$35\!\cdots\!08$$ T + $$21\!\cdots\!68$$ p T^{2} - $$67\!\cdots\!56$$ p^{2} T^{3} + $$20\!\cdots\!70$$ p^{3} T^{4} - $$67\!\cdots\!56$$ p^{75} T^{5} + $$21\!\cdots\!68$$ p^{147} T^{6} - $$35\!\cdots\!08$$ p^{219} T^{7} + p^{292} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - $$24\!\cdots\!72$$ T + $$50\!\cdots\!76$$ p T^{2} + $$49\!\cdots\!20$$ p^{2} T^{3} + $$17\!\cdots\!42$$ p^{3} T^{4} + $$49\!\cdots\!20$$ p^{75} T^{5} + $$50\!\cdots\!76$$ p^{147} T^{6} - $$24\!\cdots\!72$$ p^{219} T^{7} + p^{292} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - $$86\!\cdots\!68$$ p T + $$10\!\cdots\!68$$ p^{2} T^{2} - $$57\!\cdots\!36$$ p^{3} T^{3} + $$41\!\cdots\!70$$ p^{4} T^{4} - $$57\!\cdots\!36$$ p^{76} T^{5} + $$10\!\cdots\!68$$ p^{148} T^{6} - $$86\!\cdots\!68$$ p^{220} T^{7} + p^{292} T^{8} $
	73	$C_2 \wr S_4$	$ 1 - $$45\!\cdots\!36$$ T + $$11\!\cdots\!68$$ T^{2} - $$18\!\cdots\!80$$ T^{3} + $$22\!\cdots\!46$$ T^{4} - $$18\!\cdots\!80$$ p^{73} T^{5} + $$11\!\cdots\!68$$ p^{146} T^{6} - $$45\!\cdots\!36$$ p^{219} T^{7} + p^{292} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - $$20\!\cdots\!40$$ T + $$10\!\cdots\!56$$ T^{2} - $$16\!\cdots\!80$$ T^{3} + $$45\!\cdots\!26$$ T^{4} - $$16\!\cdots\!80$$ p^{73} T^{5} + $$10\!\cdots\!56$$ p^{146} T^{6} - $$20\!\cdots\!40$$ p^{219} T^{7} + p^{292} T^{8} $
	83	$C_2 \wr S_4$	$ 1 - $$15\!\cdots\!16$$ T + $$44\!\cdots\!48$$ T^{2} - $$49\!\cdots\!80$$ T^{3} + $$77\!\cdots\!66$$ T^{4} - $$49\!\cdots\!80$$ p^{73} T^{5} + $$44\!\cdots\!48$$ p^{146} T^{6} - $$15\!\cdots\!16$$ p^{219} T^{7} + p^{292} T^{8} $
	89	$C_2 \wr S_4$	$ 1 + $$24\!\cdots\!40$$ T + $$87\!\cdots\!76$$ T^{2} + $$14\!\cdots\!80$$ T^{3} + $$27\!\cdots\!66$$ T^{4} + $$14\!\cdots\!80$$ p^{73} T^{5} + $$87\!\cdots\!76$$ p^{146} T^{6} + $$24\!\cdots\!40$$ p^{219} T^{7} + p^{292} T^{8} $
	97	$C_2 \wr S_4$	$ 1 - $$58\!\cdots\!32$$ T + $$30\!\cdots\!92$$ T^{2} - $$13\!\cdots\!40$$ T^{3} + $$51\!\cdots\!06$$ T^{4} - $$13\!\cdots\!40$$ p^{73} T^{5} + $$30\!\cdots\!92$$ p^{146} T^{6} - $$58\!\cdots\!32$$ p^{219} T^{7} + p^{292} T^{8} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.321311394281575963670467388630, −8.306470530738937799334742129410, −7.993414950234730230019865689631, −7.889789903966694908870594716120, −7.73392643681083059820687689304, −7.24522213843305328290322266895, −6.35118098446255115275391162346, −6.31790279441654167364394774446, −6.30504111619821142827363168521, −5.41818531829205162311391076821, −5.28773759088744606405104235771, −5.15272120591769526639307173657, −4.54681436659644297667605696538, −4.25904782534659131125336323960, −3.63664937890014308213176451388, −3.63306938722294399388880299348, −3.44695335417012824069249004137, −2.81901243447341161578037480170, −2.81366920948306755492264777812, −2.20392793858434154606317959186, −2.02070197058076735062048504161, −1.71203247275573186349192304432, −1.08893243391484536301450751520, −0.67420901992411540894440151660, −0.65931688450511965720780678326, 0.65931688450511965720780678326, 0.67420901992411540894440151660, 1.08893243391484536301450751520, 1.71203247275573186349192304432, 2.02070197058076735062048504161, 2.20392793858434154606317959186, 2.81366920948306755492264777812, 2.81901243447341161578037480170, 3.44695335417012824069249004137, 3.63306938722294399388880299348, 3.63664937890014308213176451388, 4.25904782534659131125336323960, 4.54681436659644297667605696538, 5.15272120591769526639307173657, 5.28773759088744606405104235771, 5.41818531829205162311391076821, 6.30504111619821142827363168521, 6.31790279441654167364394774446, 6.35118098446255115275391162346, 7.24522213843305328290322266895, 7.73392643681083059820687689304, 7.889789903966694908870594716120, 7.993414950234730230019865689631, 8.306470530738937799334742129410, 9.321311394281575963670467388630

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 $\F_{q}$

\(L(37)\)	\(\approx\)	\(154.8346583\)
\(L(\frac12)\)	\(\approx\)	\(154.8346583\)
\(L(\frac{75}{2})\)		not available
\(L(1)\)		not available

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L-functions

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