LMFDB - L-function 8-50e4-1.1-c21e4-0-2

Dirichlet series

L(s) = 1

− 4.09e3·2^-s − 9.67e4·3^-s + 1.04e7·4^-s + 3.96e8·6^-s − 1.62e8·7^-s − 2.14e10·8^-s − 4.43e9·9^-s + 8.75e10·11^-s − 1.01e12·12^-s + 9.80e10·13^-s + 6.66e11·14^-s + 3.84e13·16^-s + 7.82e11·17^-s + 1.81e13·18^-s + 8.35e13·19^-s + 1.57e13·21^-s − 3.58e14·22^-s − 9.46e13·23^-s + 2.07e15·24^-s − 4.01e14·26^-s + 9.89e14·27^-s − 1.70e15·28^-s − 3.82e14·29^-s + 4.99e15·31^-s − 6.30e16·32^-s − 8.47e15·33^-s − 3.20e15·34^-s + ⋯

L(s) = 1

− 2.82·2^-s − 0.946·3^-s + 5·4^-s + 2.67·6^-s − 0.217·7^-s − 7.07·8^-s − 0.423·9^-s + 1.01·11^-s − 4.73·12^-s + 0.197·13^-s + 0.615·14^-s + 35/4·16^-s + 0.0940·17^-s + 1.19·18^-s + 3.12·19^-s + 0.206·21^-s − 2.87·22^-s − 0.476·23^-s + 6.68·24^-s − 0.558·26^-s + 0.925·27^-s − 1.08·28^-s − 0.168·29^-s + 1.09·31^-s − 9.89·32^-s − 0.962·33^-s − 0.266·34^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$6250000$ = $2^{4} \cdot 5^{8}$
Sign:	$1$
Analytic conductor:	$3.81299\times 10^{8}$
Root analytic conductor:	$11.8211$
Motivic weight:	$21$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 6250000,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)$

Particular Values

$L(11)$	$\approx$	$0.2387927121$
$L(\frac12)$	$\approx$	$0.2387927121$
$L(\frac{23}{2})$		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	2	$C_1$	$ ( 1 + p^{10} T )^{4} $
	5		$ 1 $
good	3	$C_2 \wr S_4$	$ 1 + 96764 T + 4598320466 p T^{2} + 9555203263360 p^{4} T^{3} + 7077208874662471 p^{8} T^{4} + 9555203263360 p^{25} T^{5} + 4598320466 p^{43} T^{6} + 96764 p^{63} T^{7} + p^{84} T^{8} $
	7	$C_2 \wr S_4$	$ 1 + 23251504 p T + 16054501545879028 p^{2} T^{2} - $$14\!\cdots\!80$$ p^{3} T^{3} + $$74\!\cdots\!86$$ p^{4} T^{4} - $$14\!\cdots\!80$$ p^{24} T^{5} + 16054501545879028 p^{44} T^{6} + 23251504 p^{64} T^{7} + p^{84} T^{8} $
	11	$C_2 \wr S_4$	$ 1 - 87559254108 T + $$25\!\cdots\!18$$ T^{2} - $$13\!\cdots\!36$$ p T^{3} + $$21\!\cdots\!95$$ p^{2} T^{4} - $$13\!\cdots\!36$$ p^{22} T^{5} + $$25\!\cdots\!18$$ p^{42} T^{6} - 87559254108 p^{63} T^{7} + p^{84} T^{8} $
	13	$C_2 \wr S_4$	$ 1 - 7546026392 p T + $$22\!\cdots\!16$$ p T^{2} + $$76\!\cdots\!40$$ p^{2} T^{3} + $$15\!\cdots\!18$$ p^{3} T^{4} + $$76\!\cdots\!40$$ p^{23} T^{5} + $$22\!\cdots\!16$$ p^{43} T^{6} - 7546026392 p^{64} T^{7} + p^{84} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - 782099305932 T + $$10\!\cdots\!06$$ p T^{2} + $$40\!\cdots\!80$$ p^{2} T^{3} + $$33\!\cdots\!47$$ p^{3} T^{4} + $$40\!\cdots\!80$$ p^{23} T^{5} + $$10\!\cdots\!06$$ p^{43} T^{6} - 782099305932 p^{63} T^{7} + p^{84} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - 83513106919940 T + $$13\!\cdots\!54$$ p T^{2} - $$32\!\cdots\!80$$ p^{2} T^{3} - $$11\!\cdots\!51$$ p^{3} T^{4} - $$32\!\cdots\!80$$ p^{23} T^{5} + $$13\!\cdots\!54$$ p^{43} T^{6} - 83513106919940 p^{63} T^{7} + p^{84} T^{8} $
	23	$C_2 \wr S_4$	$ 1 + 94696159973784 T + $$35\!\cdots\!88$$ T^{2} - $$39\!\cdots\!20$$ p T^{3} + $$16\!\cdots\!46$$ T^{4} - $$39\!\cdots\!20$$ p^{22} T^{5} + $$35\!\cdots\!88$$ p^{42} T^{6} + 94696159973784 p^{63} T^{7} + p^{84} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + 382619177097960 T + $$14\!\cdots\!16$$ T^{2} - $$31\!\cdots\!80$$ T^{3} + $$89\!\cdots\!46$$ T^{4} - $$31\!\cdots\!80$$ p^{21} T^{5} + $$14\!\cdots\!16$$ p^{42} T^{6} + 382619177097960 p^{63} T^{7} + p^{84} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - 4999617602313848 T + $$46\!\cdots\!08$$ p^{2} T^{2} - $$13\!\cdots\!76$$ T^{3} + $$10\!\cdots\!70$$ T^{4} - $$13\!\cdots\!76$$ p^{21} T^{5} + $$46\!\cdots\!08$$ p^{44} T^{6} - 4999617602313848 p^{63} T^{7} + p^{84} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - 73805465574245312 T + $$43\!\cdots\!52$$ T^{2} - $$15\!\cdots\!40$$ T^{3} + $$52\!\cdots\!66$$ T^{4} - $$15\!\cdots\!40$$ p^{21} T^{5} + $$43\!\cdots\!52$$ p^{42} T^{6} - 73805465574245312 p^{63} T^{7} + p^{84} T^{8} $
	41	$C_2 \wr S_4$	$ 1 + 67775554548131052 T + $$13\!\cdots\!78$$ T^{2} + $$11\!\cdots\!84$$ T^{3} + $$14\!\cdots\!95$$ T^{4} + $$11\!\cdots\!84$$ p^{21} T^{5} + $$13\!\cdots\!78$$ p^{42} T^{6} + 67775554548131052 p^{63} T^{7} + p^{84} T^{8} $
	43	$C_2 \wr S_4$	$ 1 + 255329082228820264 T + $$16\!\cdots\!56$$ p T^{2} + $$86\!\cdots\!40$$ T^{3} + $$15\!\cdots\!26$$ T^{4} + $$86\!\cdots\!40$$ p^{21} T^{5} + $$16\!\cdots\!56$$ p^{43} T^{6} + 255329082228820264 p^{63} T^{7} + p^{84} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + 238648375218379368 T + $$45\!\cdots\!72$$ T^{2} + $$80\!\cdots\!40$$ T^{3} + $$85\!\cdots\!46$$ T^{4} + $$80\!\cdots\!40$$ p^{21} T^{5} + $$45\!\cdots\!72$$ p^{42} T^{6} + 238648375218379368 p^{63} T^{7} + p^{84} T^{8} $
	53	$C_2 \wr S_4$	$ 1 + 1937756418161643144 T + $$10\!\cdots\!96$$ p T^{2} + $$59\!\cdots\!20$$ T^{3} + $$10\!\cdots\!26$$ T^{4} + $$59\!\cdots\!20$$ p^{21} T^{5} + $$10\!\cdots\!96$$ p^{43} T^{6} + 1937756418161643144 p^{63} T^{7} + p^{84} T^{8} $
	59	$C_2 \wr S_4$	$ 1 + 2347189971169366920 T + $$24\!\cdots\!36$$ T^{2} - $$22\!\cdots\!60$$ T^{3} + $$18\!\cdots\!86$$ T^{4} - $$22\!\cdots\!60$$ p^{21} T^{5} + $$24\!\cdots\!36$$ p^{42} T^{6} + 2347189971169366920 p^{63} T^{7} + p^{84} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - 5152202640066256688 T + $$94\!\cdots\!48$$ T^{2} - $$29\!\cdots\!96$$ T^{3} + $$37\!\cdots\!70$$ T^{4} - $$29\!\cdots\!96$$ p^{21} T^{5} + $$94\!\cdots\!48$$ p^{42} T^{6} - 5152202640066256688 p^{63} T^{7} + p^{84} T^{8} $
	67	$C_2 \wr S_4$	$ 1 + 56355797996546572108 T + $$18\!\cdots\!42$$ T^{2} + $$42\!\cdots\!40$$ T^{3} + $$73\!\cdots\!91$$ T^{4} + $$42\!\cdots\!40$$ p^{21} T^{5} + $$18\!\cdots\!42$$ p^{42} T^{6} + 56355797996546572108 p^{63} T^{7} + p^{84} T^{8} $
	71	$C_2 \wr S_4$	$ 1 + 6450631418808135672 T + $$61\!\cdots\!28$$ T^{2} + $$97\!\cdots\!64$$ T^{3} + $$10\!\cdots\!70$$ T^{4} + $$97\!\cdots\!64$$ p^{21} T^{5} + $$61\!\cdots\!28$$ p^{42} T^{6} + 6450631418808135672 p^{63} T^{7} + p^{84} T^{8} $
	73	$C_2 \wr S_4$	$ 1 + 38845779157081831564 T + $$45\!\cdots\!78$$ T^{2} + $$14\!\cdots\!00$$ T^{3} + $$86\!\cdots\!91$$ T^{4} + $$14\!\cdots\!00$$ p^{21} T^{5} + $$45\!\cdots\!78$$ p^{42} T^{6} + 38845779157081831564 p^{63} T^{7} + p^{84} T^{8} $
	79	$C_2 \wr S_4$	$ 1 + 45767420197709518720 T + $$92\!\cdots\!16$$ T^{2} - $$39\!\cdots\!60$$ T^{3} + $$11\!\cdots\!46$$ T^{4} - $$39\!\cdots\!60$$ p^{21} T^{5} + $$92\!\cdots\!16$$ p^{42} T^{6} + 45767420197709518720 p^{63} T^{7} + p^{84} T^{8} $
	83	$C_2 \wr S_4$	$ 1 + $$49\!\cdots\!64$$ T + $$14\!\cdots\!18$$ T^{2} + $$28\!\cdots\!20$$ T^{3} + $$45\!\cdots\!71$$ T^{4} + $$28\!\cdots\!20$$ p^{21} T^{5} + $$14\!\cdots\!18$$ p^{42} T^{6} + $$49\!\cdots\!64$$ p^{63} T^{7} + p^{84} T^{8} $
	89	$C_2 \wr S_4$	$ 1 + $$39\!\cdots\!40$$ T + $$21\!\cdots\!06$$ T^{2} + $$69\!\cdots\!80$$ T^{3} + $$27\!\cdots\!51$$ T^{4} + $$69\!\cdots\!80$$ p^{21} T^{5} + $$21\!\cdots\!06$$ p^{42} T^{6} + $$39\!\cdots\!40$$ p^{63} T^{7} + p^{84} T^{8} $
	97	$C_2 \wr S_4$	$ 1 + $$94\!\cdots\!08$$ T + $$18\!\cdots\!12$$ T^{2} + $$12\!\cdots\!60$$ T^{3} + $$13\!\cdots\!26$$ T^{4} + $$12\!\cdots\!60$$ p^{21} T^{5} + $$18\!\cdots\!12$$ p^{42} T^{6} + $$94\!\cdots\!08$$ 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

−7.61163298235715700059311702932, −7.43330836836447757081034532858, −6.90022787906107811904833961560, −6.89457814904450718058354929692, −6.53525016400548429121198003217, −6.21570973424463301163526919381, −5.83291957898879035963299413351, −5.66155733786185090493260809070, −5.60443540839528690384873527421, −4.92686807538430127654484206633, −4.64250606391870930317138077826, −4.25555708082972646429059387951, −3.86531803198152086218551593765, −3.05575751006884189435176829278, −3.00854263888057491497167969495, −2.97277030419882054142181476157, −2.97164834377977849841012288512, −1.92275602464266773817280822177, −1.85403423575163878546077054850, −1.40154725574623696243498297341, −1.34821863283083695095565571867, −0.930385705593866210752909475586, −0.901088590241127066286705778149, −0.22561232009187428999847641153, −0.21850794754384001389475704156, 0.21850794754384001389475704156, 0.22561232009187428999847641153, 0.901088590241127066286705778149, 0.930385705593866210752909475586, 1.34821863283083695095565571867, 1.40154725574623696243498297341, 1.85403423575163878546077054850, 1.92275602464266773817280822177, 2.97164834377977849841012288512, 2.97277030419882054142181476157, 3.00854263888057491497167969495, 3.05575751006884189435176829278, 3.86531803198152086218551593765, 4.25555708082972646429059387951, 4.64250606391870930317138077826, 4.92686807538430127654484206633, 5.60443540839528690384873527421, 5.66155733786185090493260809070, 5.83291957898879035963299413351, 6.21570973424463301163526919381, 6.53525016400548429121198003217, 6.89457814904450718058354929692, 6.90022787906107811904833961560, 7.43330836836447757081034532858, 7.61163298235715700059311702932

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(11)\)	\(\approx\)	\(0.2387927121\)
\(L(\frac12)\)	\(\approx\)	\(0.2387927121\)
\(L(\frac{23}{2})\)		not available
\(L(1)\)		not available

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