LMFDB - L-function 4-507e2-1.1-c0e2-0-0

Dirichlet series

L(s) = 1

+ 3^-s − 4^-s − 12^-s + 2·25^-s − 27^-s + 2·43^-s + 49^-s + 2·61^-s + 64^-s + 2·75^-s − 4·79^-s − 81^-s − 2·100^-s − 4·103^-s + 108^-s − 121^-s + 127^-s + 2·129^-s + 131^-s + 137^-s + 139^-s + 147^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$257049$ = $3^{2} \cdot 13^{4}$
Sign:	$1$
Analytic conductor:	$0.0640221$
Root analytic conductor:	$0.503016$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 257049,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8074079058$
$L(\frac12)$	$\approx$	$0.8074079058$
$L(1)$		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	$C_2$	$ 1 - T + T^{2} $
	13		$ 1 $
good	2	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	5	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	7	$C_2^2$	$ 1 - T^{2} + T^{4} $
	11	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	17	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	19	$C_2^2$	$ 1 - T^{2} + T^{4} $
	23	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	29	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	31	$C_2$	$ ( 1 + T^{2} )^{2} $
	37	$C_2^2$	$ 1 - T^{2} + T^{4} $
	41	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	43	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	47	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	53	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	61	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	67	$C_2^2$	$ 1 - T^{2} + T^{4} $
	71	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	73	$C_2$	$ ( 1 + T^{2} )^{2} $
	79	$C_1$	$ ( 1 + T )^{4} $
	83	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	89	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	97	$C_2^2$	$ 1 - T^{2} + T^{4} $
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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

−11.23643677776475616866431726563, −10.88136809221007974248749099302, −10.38213482327925293701242344334, −9.898100540943651218271153597092, −9.395251441063559319166592185572, −9.047730148575011384562310254307, −8.764718951248357688220939616893, −8.346696330881788513305554161941, −7.989647788109809637648342374004, −7.22915406559593898589107009644, −7.02182683090080460150910108269, −6.33674402936279594608132512011, −5.46862914152788743188557005250, −5.41230366713438299869784166266, −4.49409891423537551158953915286, −4.15785030189895747729642588146, −3.62455441534759852307591575842, −2.68427045924795299142405703463, −2.58616032024483267256904783173, −1.24681300363605832322506620524, 1.24681300363605832322506620524, 2.58616032024483267256904783173, 2.68427045924795299142405703463, 3.62455441534759852307591575842, 4.15785030189895747729642588146, 4.49409891423537551158953915286, 5.41230366713438299869784166266, 5.46862914152788743188557005250, 6.33674402936279594608132512011, 7.02182683090080460150910108269, 7.22915406559593898589107009644, 7.989647788109809637648342374004, 8.346696330881788513305554161941, 8.764718951248357688220939616893, 9.047730148575011384562310254307, 9.395251441063559319166592185572, 9.898100540943651218271153597092, 10.38213482327925293701242344334, 10.88136809221007974248749099302, 11.23643677776475616866431726563

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Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.8074079058\)
\(L(\frac12)\)	\(\approx\)	\(0.8074079058\)
\(L(1)\)		not available
\(L(1)\)		not available

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