LMFDB - L-function 8-45e4-1.1-c21e4-0-0

Dirichlet series

L(s) = 1

+ 897·2^-s − 4.37e6·4^-s − 3.90e7·5^-s − 2.34e8·7^-s − 4.81e9·8^-s − 3.50e10·10^-s − 3.14e10·11^-s − 2.70e10·13^-s − 2.10e11·14^-s + 6.70e12·16^-s + 2.94e12·17^-s − 2.42e13·19^-s + 1.70e14·20^-s − 2.82e13·22^-s − 1.03e13·23^-s + 9.53e14·25^-s − 2.42e13·26^-s + 1.02e15·28^-s − 4.72e15·29^-s − 1.09e15·31^-s + 8.96e15·32^-s + 2.64e15·34^-s + 9.16e15·35^-s + 4.81e15·37^-s − 2.17e16·38^-s + 1.88e17·40^-s − 2.89e17·41^-s + ⋯

L(s) = 1

+ 0.619·2^-s − 2.08·4^-s − 1.78·5^-s − 0.313·7^-s − 1.58·8^-s − 1.10·10^-s − 0.366·11^-s − 0.0543·13^-s − 0.194·14^-s + 1.52·16^-s + 0.354·17^-s − 0.908·19^-s + 3.73·20^-s − 0.226·22^-s − 0.0520·23^-s + 2·25^-s − 0.0336·26^-s + 0.654·28^-s − 2.08·29^-s − 0.239·31^-s + 1.40·32^-s + 0.219·34^-s + 0.561·35^-s + 0.164·37^-s − 0.562·38^-s + 2.83·40^-s − 3.37·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$4100625$ = $3^{8} \cdot 5^{4}$
Sign:	$1$
Analytic conductor:	$2.50170\times 10^{8}$
Root analytic conductor:	$11.2144$
Motivic weight:	$21$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 4100625,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)$

Particular Values

$L(11)$	$\approx$	$0.01920876667$
$L(\frac12)$	$\approx$	$0.01920876667$
$L(\frac{23}{2})$		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	3		$ 1 $
	5	$C_1$	$ ( 1 + p^{10} T )^{4} $
good	2	$C_2 \wr S_4$	$ 1 - 897 T + 2589085 p T^{2} - 117236669 p^{5} T^{3} + 14637391839 p^{10} T^{4} - 117236669 p^{26} T^{5} + 2589085 p^{43} T^{6} - 897 p^{63} T^{7} + p^{84} T^{8} $
	7	$C_2 \wr S_4$	$ 1 + 4787296 p^{2} T + 7592135803176028 p^{2} T^{2} - $$13\!\cdots\!36$$ p^{4} T^{3} + $$73\!\cdots\!10$$ p^{4} T^{4} - $$13\!\cdots\!36$$ p^{25} T^{5} + 7592135803176028 p^{44} T^{6} + 4787296 p^{65} T^{7} + p^{84} T^{8} $
	11	$C_2 \wr S_4$	$ 1 + 31491830256 T + $$93\!\cdots\!92$$ p T^{2} + $$46\!\cdots\!00$$ p^{2} T^{3} + $$53\!\cdots\!06$$ p^{3} T^{4} + $$46\!\cdots\!00$$ p^{23} T^{5} + $$93\!\cdots\!92$$ p^{43} T^{6} + 31491830256 p^{63} T^{7} + p^{84} T^{8} $
	13	$C_2 \wr S_4$	$ 1 + 27017977768 T - $$92\!\cdots\!72$$ p T^{2} + $$33\!\cdots\!24$$ p^{2} T^{3} + $$39\!\cdots\!30$$ p^{3} T^{4} + $$33\!\cdots\!24$$ p^{23} T^{5} - $$92\!\cdots\!72$$ p^{43} T^{6} + 27017977768 p^{63} T^{7} + p^{84} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - 2946095028888 T + $$67\!\cdots\!16$$ p T^{2} - $$25\!\cdots\!20$$ p^{2} T^{3} + $$19\!\cdots\!02$$ p^{3} T^{4} - $$25\!\cdots\!20$$ p^{23} T^{5} + $$67\!\cdots\!16$$ p^{43} T^{6} - 2946095028888 p^{63} T^{7} + p^{84} T^{8} $
	19	$C_2 \wr S_4$	$ 1 + 24270353300752 T + $$48\!\cdots\!92$$ p^{2} T^{2} + $$36\!\cdots\!36$$ p^{3} T^{3} + $$21\!\cdots\!26$$ p^{3} T^{4} + $$36\!\cdots\!36$$ p^{24} T^{5} + $$48\!\cdots\!92$$ p^{44} T^{6} + 24270353300752 p^{63} T^{7} + p^{84} T^{8} $
	23	$C_2 \wr S_4$	$ 1 + 10350924920928 T + $$70\!\cdots\!08$$ T^{2} + $$70\!\cdots\!12$$ T^{3} + $$23\!\cdots\!10$$ T^{4} + $$70\!\cdots\!12$$ p^{21} T^{5} + $$70\!\cdots\!08$$ p^{42} T^{6} + 10350924920928 p^{63} T^{7} + p^{84} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + 4728924677079096 T + $$11\!\cdots\!00$$ T^{2} + $$17\!\cdots\!92$$ T^{3} + $$17\!\cdots\!18$$ T^{4} + $$17\!\cdots\!92$$ p^{21} T^{5} + $$11\!\cdots\!00$$ p^{42} T^{6} + 4728924677079096 p^{63} T^{7} + p^{84} T^{8} $
	31	$C_2 \wr S_4$	$ 1 + 1094923910405536 T + $$68\!\cdots\!48$$ T^{2} + $$44\!\cdots\!48$$ T^{3} + $$19\!\cdots\!54$$ T^{4} + $$44\!\cdots\!48$$ p^{21} T^{5} + $$68\!\cdots\!48$$ p^{42} T^{6} + 1094923910405536 p^{63} T^{7} + p^{84} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - 4813435696247096 T + $$17\!\cdots\!12$$ T^{2} - $$33\!\cdots\!36$$ T^{3} + $$15\!\cdots\!50$$ T^{4} - $$33\!\cdots\!36$$ p^{21} T^{5} + $$17\!\cdots\!12$$ p^{42} T^{6} - 4813435696247096 p^{63} T^{7} + p^{84} T^{8} $
	41	$C_2 \wr S_4$	$ 1 + 289731591445930344 T + $$49\!\cdots\!68$$ T^{2} + $$57\!\cdots\!52$$ T^{3} + $$55\!\cdots\!14$$ T^{4} + $$57\!\cdots\!52$$ p^{21} T^{5} + $$49\!\cdots\!68$$ p^{42} T^{6} + 289731591445930344 p^{63} T^{7} + p^{84} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - 451091912658458000 T + $$13\!\cdots\!00$$ T^{2} - $$27\!\cdots\!00$$ T^{3} + $$44\!\cdots\!98$$ T^{4} - $$27\!\cdots\!00$$ p^{21} T^{5} + $$13\!\cdots\!00$$ p^{42} T^{6} - 451091912658458000 p^{63} T^{7} + p^{84} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + 813883435638492480 T + $$59\!\cdots\!20$$ T^{2} + $$27\!\cdots\!60$$ T^{3} + $$11\!\cdots\!18$$ T^{4} + $$27\!\cdots\!60$$ p^{21} T^{5} + $$59\!\cdots\!20$$ p^{42} T^{6} + 813883435638492480 p^{63} T^{7} + p^{84} T^{8} $
	53	$C_2 \wr S_4$	$ 1 + 697278335404085208 T + $$47\!\cdots\!28$$ T^{2} + $$22\!\cdots\!72$$ T^{3} + $$10\!\cdots\!50$$ T^{4} + $$22\!\cdots\!72$$ p^{21} T^{5} + $$47\!\cdots\!28$$ p^{42} T^{6} + 697278335404085208 p^{63} T^{7} + p^{84} T^{8} $
	59	$C_2 \wr S_4$	$ 1 + 6622888614569598192 T + $$40\!\cdots\!72$$ T^{2} + $$20\!\cdots\!04$$ T^{3} + $$88\!\cdots\!34$$ T^{4} + $$20\!\cdots\!04$$ p^{21} T^{5} + $$40\!\cdots\!72$$ p^{42} T^{6} + 6622888614569598192 p^{63} T^{7} + p^{84} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - 7390887218011683320 T + $$30\!\cdots\!96$$ T^{2} - $$27\!\cdots\!80$$ T^{3} + $$27\!\cdots\!46$$ T^{4} - $$27\!\cdots\!80$$ p^{21} T^{5} + $$30\!\cdots\!96$$ p^{42} T^{6} - 7390887218011683320 p^{63} T^{7} + p^{84} T^{8} $
	67	$C_2 \wr S_4$	$ 1 + 24188188449376788688 T + $$66\!\cdots\!72$$ T^{2} + $$12\!\cdots\!80$$ T^{3} + $$22\!\cdots\!26$$ T^{4} + $$12\!\cdots\!80$$ p^{21} T^{5} + $$66\!\cdots\!72$$ p^{42} T^{6} + 24188188449376788688 p^{63} T^{7} + p^{84} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - 37390337803999713312 T + $$18\!\cdots\!88$$ T^{2} - $$49\!\cdots\!64$$ T^{3} + $$15\!\cdots\!70$$ T^{4} - $$49\!\cdots\!64$$ p^{21} T^{5} + $$18\!\cdots\!88$$ p^{42} T^{6} - 37390337803999713312 p^{63} T^{7} + p^{84} T^{8} $
	73	$C_2 \wr S_4$	$ 1 + 37253672904265201432 T + $$53\!\cdots\!28$$ T^{2} + $$13\!\cdots\!88$$ T^{3} + $$10\!\cdots\!30$$ T^{4} + $$13\!\cdots\!88$$ p^{21} T^{5} + $$53\!\cdots\!28$$ p^{42} T^{6} + 37253672904265201432 p^{63} T^{7} + p^{84} T^{8} $
	79	$C_2 \wr S_4$	$ 1 + $$10\!\cdots\!00$$ T + $$20\!\cdots\!16$$ T^{2} + $$13\!\cdots\!00$$ T^{3} + $$17\!\cdots\!46$$ T^{4} + $$13\!\cdots\!00$$ p^{21} T^{5} + $$20\!\cdots\!16$$ p^{42} T^{6} + $$10\!\cdots\!00$$ p^{63} T^{7} + p^{84} T^{8} $
	83	$C_2 \wr S_4$	$ 1 - $$14\!\cdots\!96$$ T + $$49\!\cdots\!60$$ T^{2} - $$54\!\cdots\!56$$ T^{3} + $$11\!\cdots\!26$$ T^{4} - $$54\!\cdots\!56$$ p^{21} T^{5} + $$49\!\cdots\!60$$ p^{42} T^{6} - $$14\!\cdots\!96$$ p^{63} T^{7} + p^{84} T^{8} $
	89	$C_2 \wr S_4$	$ 1 + $$55\!\cdots\!48$$ T + $$43\!\cdots\!52$$ T^{2} + $$14\!\cdots\!96$$ T^{3} + $$60\!\cdots\!14$$ T^{4} + $$14\!\cdots\!96$$ p^{21} T^{5} + $$43\!\cdots\!52$$ p^{42} T^{6} + $$55\!\cdots\!48$$ p^{63} T^{7} + p^{84} T^{8} $
	97	$C_2 \wr S_4$	$ 1 + $$23\!\cdots\!28$$ T + $$13\!\cdots\!32$$ T^{2} + $$61\!\cdots\!20$$ T^{3} + $$80\!\cdots\!66$$ T^{4} + $$61\!\cdots\!20$$ p^{21} T^{5} + $$13\!\cdots\!32$$ p^{42} T^{6} + $$23\!\cdots\!28$$ p^{63} T^{7} + p^{84} 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

−7.925661808048971603484989154936, −7.47400900377655106709746858276, −7.11931413950061897356061873488, −6.71548747807764158673202294660, −6.69661646011760205096314348576, −6.08379063346739706529498170478, −5.49282207059025500609540872302, −5.48762596346637760272339649288, −5.37802426682904202908343187039, −4.64948348735575554198424198864, −4.55586248848710531257708936492, −4.28034506390677916071463978657, −4.25595363719694055877070508889, −3.83725941632738691745049331827, −3.34414659808914850725861468710, −3.21115415125424645186078196694, −3.20942321412360675521358032589, −2.63281472626392836416860681181, −1.98420057947417468079924546350, −1.67414911027982636189640623942, −1.63762515646515751208863686285, −0.838567829295619169433666960052, −0.50659759460686242462831000900, −0.44074888670186003444212388732, −0.03115564611230382885270971954, 0.03115564611230382885270971954, 0.44074888670186003444212388732, 0.50659759460686242462831000900, 0.838567829295619169433666960052, 1.63762515646515751208863686285, 1.67414911027982636189640623942, 1.98420057947417468079924546350, 2.63281472626392836416860681181, 3.20942321412360675521358032589, 3.21115415125424645186078196694, 3.34414659808914850725861468710, 3.83725941632738691745049331827, 4.25595363719694055877070508889, 4.28034506390677916071463978657, 4.55586248848710531257708936492, 4.64948348735575554198424198864, 5.37802426682904202908343187039, 5.48762596346637760272339649288, 5.49282207059025500609540872302, 6.08379063346739706529498170478, 6.69661646011760205096314348576, 6.71548747807764158673202294660, 7.11931413950061897356061873488, 7.47400900377655106709746858276, 7.925661808048971603484989154936

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 $\F_{q}$

\(L(11)\)	\(\approx\)	\(0.01920876667\)
\(L(\frac12)\)	\(\approx\)	\(0.01920876667\)
\(L(\frac{23}{2})\)		not available
\(L(1)\)		not available

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