LMFDB - L-function 4-104e2-1.1-c1e2-0-2

Dirichlet series

L(s) = 1

− 2·2^-s + 2·4^-s + 6·7^-s + 5·9^-s − 12·14^-s − 4·16^-s − 14·17^-s − 10·18^-s + 8·23^-s + 25^-s + 12·28^-s − 16·31^-s + 8·32^-s + 28·34^-s + 10·36^-s + 4·41^-s − 16·46^-s − 14·47^-s + 13·49^-s − 2·50^-s + 32·62^-s + 30·63^-s − 8·64^-s − 28·68^-s − 6·71^-s + 28·73^-s − 20·79^-s + ⋯

L(s) = 1

− 1.41·2^-s + 4^-s + 2.26·7^-s + 5/3·9^-s − 3.20·14^-s − 16^-s − 3.39·17^-s − 2.35·18^-s + 1.66·23^-s + 1/5·25^-s + 2.26·28^-s − 2.87·31^-s + 1.41·32^-s + 4.80·34^-s + 5/3·36^-s + 0.624·41^-s − 2.35·46^-s − 2.04·47^-s + 13/7·49^-s − 0.282·50^-s + 4.06·62^-s + 3.77·63^-s − 64^-s − 3.39·68^-s − 0.712·71^-s + 3.27·73^-s − 2.25·79^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$10816$ = $2^{6} \cdot 13^{2}$
Sign:	$1$
Analytic conductor:	$0.689637$
Root analytic conductor:	$0.911287$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 10816,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.6630077108$
$L(\frac12)$	$\approx$	$0.6630077108$
$L(\frac{3}{2})$		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	$C_2$	$ 1 + p T + p T^{2} $
	13	$C_2$	$ 1 + T^{2} $
good	3	$C_2^2$	$ 1 - 5 T^{2} + p^{2} T^{4} $
	5	$C_2^2$	$ 1 - T^{2} + p^{2} T^{4} $
	7	$C_2$	$ ( 1 - 3 T + p T^{2} )^{2} $
	11	$C_2$	$ ( 1 - p T^{2} )^{2} $
	17	$C_2$	$ ( 1 + 7 T + p T^{2} )^{2} $
	19	$C_2^2$	$ 1 - 22 T^{2} + p^{2} T^{4} $
	23	$C_2$	$ ( 1 - 4 T + p T^{2} )^{2} $
	29	$C_2$	$ ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) $
	31	$C_2$	$ ( 1 + 8 T + p T^{2} )^{2} $
	37	$C_2^2$	$ 1 - 25 T^{2} + p^{2} T^{4} $
	41	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	43	$C_2^2$	$ 1 - 85 T^{2} + p^{2} T^{4} $
	47	$C_2$	$ ( 1 + 7 T + p T^{2} )^{2} $
	53	$C_2$	$ ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) $
	59	$C_2^2$	$ 1 + 78 T^{2} + p^{2} T^{4} $
	61	$C_2$	$ ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) $
	67	$C_2^2$	$ 1 - 130 T^{2} + p^{2} T^{4} $
	71	$C_2$	$ ( 1 + 3 T + p T^{2} )^{2} $
	73	$C_2$	$ ( 1 - 14 T + p T^{2} )^{2} $
	79	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
	83	$C_2^2$	$ 1 + 30 T^{2} + p^{2} T^{4} $
	89	$C_2$	$ ( 1 + p T^{2} )^{2} $
	97	$C_2$	$ ( 1 - 8 T + p T^{2} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−14.15440631000278160403879527824, −13.20852039438558746320617634653, −13.19648486565889657619060078782, −12.58600300582480593983015104864, −11.34323865242704371941958808037, −11.26399418836677676300121220421, −10.92856591718551450502520099306, −10.52597930620038220776880637397, −9.508812300858634453042338551359, −9.199662015815668273650387336959, −8.626171664997409426871552911051, −8.195138470375305827482347337470, −7.40053747850704823055861792041, −7.10015539971666652880095457705, −6.57035023944590742165974568824, −5.01634651797727751766763907520, −4.73879432561462994763240805629, −4.10530874863142447259873660113, −2.05759140072052388639569224374, −1.61690266206879990138128123661, 1.61690266206879990138128123661, 2.05759140072052388639569224374, 4.10530874863142447259873660113, 4.73879432561462994763240805629, 5.01634651797727751766763907520, 6.57035023944590742165974568824, 7.10015539971666652880095457705, 7.40053747850704823055861792041, 8.195138470375305827482347337470, 8.626171664997409426871552911051, 9.199662015815668273650387336959, 9.508812300858634453042338551359, 10.52597930620038220776880637397, 10.92856591718551450502520099306, 11.26399418836677676300121220421, 11.34323865242704371941958808037, 12.58600300582480593983015104864, 13.19648486565889657619060078782, 13.20852039438558746320617634653, 14.15440631000278160403879527824

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6630077108\)
\(L(\frac12)\)	\(\approx\)	\(0.6630077108\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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