LMFDB - L-function 8-78e4-1.1-c15e4-0-3

Dirichlet series

L(s) = 1

+ 512·2^-s + 8.74e3·3^-s + 1.63e5·4^-s + 3.00e5·5^-s + 4.47e6·6^-s + 2.89e6·7^-s + 4.19e7·8^-s + 4.78e7·9^-s + 1.53e8·10^-s + 1.23e8·11^-s + 1.43e9·12^-s − 2.50e8·13^-s + 1.48e9·14^-s + 2.62e9·15^-s + 9.39e9·16^-s + 4.54e9·17^-s + 2.44e10·18^-s + 3.53e9·19^-s + 4.91e10·20^-s + 2.53e10·21^-s + 6.32e10·22^-s + 4.77e8·23^-s + 3.66e11·24^-s + 6.69e9·25^-s − 1.28e11·26^-s + 2.09e11·27^-s + 4.74e11·28^-s + ⋯

L(s) = 1

+ 2.82·2^-s + 2.30·3^-s + 5·4^-s + 1.71·5^-s + 6.53·6^-s + 1.32·7^-s + 7.07·8^-s + 10/3·9^-s + 4.86·10^-s + 1.91·11^-s + 11.5·12^-s − 1.10·13^-s + 3.75·14^-s + 3.96·15^-s + 35/4·16^-s + 2.68·17^-s + 9.42·18^-s + 0.906·19^-s + 8.59·20^-s + 3.06·21^-s + 5.40·22^-s + 0.0292·23^-s + 16.3·24^-s + 0.219·25^-s − 3.13·26^-s + 3.84·27^-s + 6.64·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$37015056$ = $2^{4} \cdot 3^{4} \cdot 13^{4}$
Sign:	$1$
Analytic conductor:	$1.53460\times 10^{8}$
Root analytic conductor:	$10.5499$
Motivic weight:	$15$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 37015056,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)$

Particular Values

$L(8)$	$\approx$	$952.3424560$
$L(\frac12)$	$\approx$	$952.3424560$
$L(\frac{17}{2})$		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^{7} T )^{4} $
	3	$C_1$	$ ( 1 - p^{7} T )^{4} $
	13	$C_1$	$ ( 1 + p^{7} T )^{4} $
good	5	$C_2 \wr S_4$	$ 1 - 60044 p T + 3337662716 p^{2} T^{2} - 131109067049876 p^{3} T^{3} + 5371991685731690086 p^{4} T^{4} - 131109067049876 p^{18} T^{5} + 3337662716 p^{32} T^{6} - 60044 p^{46} T^{7} + p^{60} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - 2894308 T + 1433276120092 p T^{2} - 147747052927927444 p^{2} T^{3} + $$74\!\cdots\!90$$ p^{3} T^{4} - 147747052927927444 p^{17} T^{5} + 1433276120092 p^{31} T^{6} - 2894308 p^{45} T^{7} + p^{60} T^{8} $
	11	$C_2 \wr S_4$	$ 1 - 123480656 T + 1267959675433852 p T^{2} - $$91\!\cdots\!20$$ p^{2} T^{3} + $$55\!\cdots\!26$$ p^{3} T^{4} - $$91\!\cdots\!20$$ p^{17} T^{5} + 1267959675433852 p^{31} T^{6} - 123480656 p^{45} T^{7} + p^{60} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - 4547579520 T + 15795958402819961420 T^{2} - $$36\!\cdots\!60$$ T^{3} + $$71\!\cdots\!98$$ T^{4} - $$36\!\cdots\!60$$ p^{15} T^{5} + 15795958402819961420 p^{30} T^{6} - 4547579520 p^{45} T^{7} + p^{60} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - 3530203884 T + 38743662870709295764 T^{2} - $$12\!\cdots\!16$$ T^{3} + $$70\!\cdots\!74$$ T^{4} - $$12\!\cdots\!16$$ p^{15} T^{5} + 38743662870709295764 p^{30} T^{6} - 3530203884 p^{45} T^{7} + p^{60} T^{8} $
	23	$C_2 \wr S_4$	$ 1 - 477775360 T + $$53\!\cdots\!28$$ T^{2} - $$68\!\cdots\!20$$ p T^{3} + $$18\!\cdots\!94$$ T^{4} - $$68\!\cdots\!20$$ p^{16} T^{5} + $$53\!\cdots\!28$$ p^{30} T^{6} - 477775360 p^{45} T^{7} + p^{60} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + 32296563560 T + $$11\!\cdots\!24$$ T^{2} + $$49\!\cdots\!60$$ T^{3} + $$17\!\cdots\!46$$ T^{4} + $$49\!\cdots\!60$$ p^{15} T^{5} + $$11\!\cdots\!24$$ p^{30} T^{6} + 32296563560 p^{45} T^{7} + p^{60} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - 302848932388 T + $$10\!\cdots\!96$$ T^{2} - $$18\!\cdots\!28$$ T^{3} + $$36\!\cdots\!74$$ T^{4} - $$18\!\cdots\!28$$ p^{15} T^{5} + $$10\!\cdots\!96$$ p^{30} T^{6} - 302848932388 p^{45} T^{7} + p^{60} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - 486884740368 T + $$86\!\cdots\!04$$ T^{2} - $$36\!\cdots\!56$$ T^{3} + $$38\!\cdots\!90$$ T^{4} - $$36\!\cdots\!56$$ p^{15} T^{5} + $$86\!\cdots\!04$$ p^{30} T^{6} - 486884740368 p^{45} T^{7} + p^{60} T^{8} $
	41	$C_2 \wr S_4$	$ 1 + 48964324452 T + $$58\!\cdots\!00$$ T^{2} + $$20\!\cdots\!76$$ T^{3} + $$13\!\cdots\!98$$ T^{4} + $$20\!\cdots\!76$$ p^{15} T^{5} + $$58\!\cdots\!00$$ p^{30} T^{6} + 48964324452 p^{45} T^{7} + p^{60} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - 781336193080 T + $$37\!\cdots\!20$$ T^{2} - $$48\!\cdots\!60$$ T^{3} + $$14\!\cdots\!98$$ T^{4} - $$48\!\cdots\!60$$ p^{15} T^{5} + $$37\!\cdots\!20$$ p^{30} T^{6} - 781336193080 p^{45} T^{7} + p^{60} T^{8} $
	47	$C_2 \wr S_4$	$ 1 - 1925546085944 T + $$21\!\cdots\!48$$ T^{2} - $$40\!\cdots\!00$$ T^{3} + $$36\!\cdots\!46$$ T^{4} - $$40\!\cdots\!00$$ p^{15} T^{5} + $$21\!\cdots\!48$$ p^{30} T^{6} - 1925546085944 p^{45} T^{7} + p^{60} T^{8} $
	53	$C_2 \wr S_4$	$ 1 - 3784002694008 T + $$18\!\cdots\!04$$ T^{2} - $$54\!\cdots\!08$$ T^{3} + $$17\!\cdots\!58$$ T^{4} - $$54\!\cdots\!08$$ p^{15} T^{5} + $$18\!\cdots\!04$$ p^{30} T^{6} - 3784002694008 p^{45} T^{7} + p^{60} T^{8} $
	59	$C_2 \wr S_4$	$ 1 + 3338828832488 T + $$11\!\cdots\!92$$ T^{2} + $$28\!\cdots\!76$$ T^{3} + $$59\!\cdots\!14$$ T^{4} + $$28\!\cdots\!76$$ p^{15} T^{5} + $$11\!\cdots\!92$$ p^{30} T^{6} + 3338828832488 p^{45} T^{7} + p^{60} T^{8} $
	61	$C_2 \wr S_4$	$ 1 + 2964321922952 T + $$48\!\cdots\!36$$ T^{2} + $$14\!\cdots\!12$$ T^{3} - $$15\!\cdots\!46$$ T^{4} + $$14\!\cdots\!12$$ p^{15} T^{5} + $$48\!\cdots\!36$$ p^{30} T^{6} + 2964321922952 p^{45} T^{7} + p^{60} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - 28476463721140 T + $$79\!\cdots\!44$$ T^{2} - $$19\!\cdots\!00$$ T^{3} + $$27\!\cdots\!82$$ T^{4} - $$19\!\cdots\!00$$ p^{15} T^{5} + $$79\!\cdots\!44$$ p^{30} T^{6} - 28476463721140 p^{45} T^{7} + p^{60} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - 40721833289696 T + $$19\!\cdots\!52$$ T^{2} - $$55\!\cdots\!00$$ T^{3} + $$16\!\cdots\!86$$ T^{4} - $$55\!\cdots\!00$$ p^{15} T^{5} + $$19\!\cdots\!52$$ p^{30} T^{6} - 40721833289696 p^{45} T^{7} + p^{60} T^{8} $
	73	$C_2 \wr S_4$	$ 1 - 90777676225904 T + $$25\!\cdots\!84$$ T^{2} - $$18\!\cdots\!88$$ T^{3} + $$31\!\cdots\!54$$ T^{4} - $$18\!\cdots\!88$$ p^{15} T^{5} + $$25\!\cdots\!84$$ p^{30} T^{6} - 90777676225904 p^{45} T^{7} + p^{60} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - 43793210668880 T + $$74\!\cdots\!88$$ T^{2} - $$12\!\cdots\!40$$ T^{3} + $$28\!\cdots\!38$$ T^{4} - $$12\!\cdots\!40$$ p^{15} T^{5} + $$74\!\cdots\!88$$ p^{30} T^{6} - 43793210668880 p^{45} T^{7} + p^{60} T^{8} $
	83	$C_2 \wr S_4$	$ 1 + 34142102730120 T + $$66\!\cdots\!56$$ T^{2} - $$19\!\cdots\!00$$ T^{3} + $$18\!\cdots\!82$$ T^{4} - $$19\!\cdots\!00$$ p^{15} T^{5} + $$66\!\cdots\!56$$ p^{30} T^{6} + 34142102730120 p^{45} T^{7} + p^{60} T^{8} $
	89	$C_2 \wr S_4$	$ 1 - 192561551798100 T + $$58\!\cdots\!24$$ T^{2} - $$74\!\cdots\!00$$ T^{3} + $$14\!\cdots\!46$$ T^{4} - $$74\!\cdots\!00$$ p^{15} T^{5} + $$58\!\cdots\!24$$ p^{30} T^{6} - 192561551798100 p^{45} T^{7} + p^{60} T^{8} $
	97	$C_2 \wr S_4$	$ 1 - 577119276333152 T + $$23\!\cdots\!88$$ T^{2} - $$10\!\cdots\!48$$ T^{3} + $$21\!\cdots\!70$$ T^{4} - $$10\!\cdots\!48$$ p^{15} T^{5} + $$23\!\cdots\!88$$ p^{30} T^{6} - 577119276333152 p^{45} T^{7} + p^{60} 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

−7.83821600155544995602493134055, −7.31676589743585453206415107014, −6.89620910602359307226958473275, −6.76655063789355511218723854544, −6.65091611667621773573853184828, −5.91342756928131190382325822551, −5.77873602570806598067135119397, −5.51923239366434395606591154225, −5.48293825271472002419364863204, −4.76259966712809704402551564132, −4.58980740756555317198257758051, −4.36482402463906523729659523721, −4.26936506399919149038119472562, −3.55565369003621944895955022835, −3.34696110330990108961612593226, −3.30066358835194944584443706158, −3.08758459761607773546666182070, −2.40999954559774128332511152000, −2.27157808430043246815240159717, −2.16387668452842279012846382040, −1.74700047165720176350616294911, −1.57141321258978140650518849150, −1.12070734963143983593703013817, −0.948908423856332618561727565687, −0.803888100446549459071084014042, 0.803888100446549459071084014042, 0.948908423856332618561727565687, 1.12070734963143983593703013817, 1.57141321258978140650518849150, 1.74700047165720176350616294911, 2.16387668452842279012846382040, 2.27157808430043246815240159717, 2.40999954559774128332511152000, 3.08758459761607773546666182070, 3.30066358835194944584443706158, 3.34696110330990108961612593226, 3.55565369003621944895955022835, 4.26936506399919149038119472562, 4.36482402463906523729659523721, 4.58980740756555317198257758051, 4.76259966712809704402551564132, 5.48293825271472002419364863204, 5.51923239366434395606591154225, 5.77873602570806598067135119397, 5.91342756928131190382325822551, 6.65091611667621773573853184828, 6.76655063789355511218723854544, 6.89620910602359307226958473275, 7.31676589743585453206415107014, 7.83821600155544995602493134055

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 $\F_{q}$

\(L(8)\)	\(\approx\)	\(952.3424560\)
\(L(\frac12)\)	\(\approx\)	\(952.3424560\)
\(L(\frac{17}{2})\)		not available
\(L(1)\)		not available

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