LMFDB - L-function 4-507e2-1.1-c0e2-0-1

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

$L(\frac{1}{2})$	$\approx$	$1.187138105$
$L(\frac12)$	$\approx$	$1.187138105$
$L(1)$		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	$C_2^2$	$ 1 - T^{2} + T^{4} $
	5	$C_2$	$ ( 1 + T^{2} )^{2} $
	7	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	11	$C_2^2$	$ 1 - T^{2} + T^{4} $
	17	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	19	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	23	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	29	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	31	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	37	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	41	$C_2^2$	$ 1 - T^{2} + T^{4} $
	43	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	47	$C_2$	$ ( 1 + T^{2} )^{2} $
	53	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	59	$C_2^2$	$ 1 - T^{2} + T^{4} $
	61	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	67	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{2} ) $
	71	$C_2^2$	$ 1 - T^{2} + T^{4} $
	73	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	79	$C_1$	$ ( 1 + T )^{4} $
	83	$C_2$	$ ( 1 + T^{2} )^{2} $
	89	$C_2^2$	$ 1 - T^{2} + T^{4} $
	97	$C_2$	$ ( 1 - T + T^{2} )( 1 + T + T^{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

−11.44895139159288995580079601776, −11.11886424873715407517357760874, −10.20845153979420460309348321019, −10.04713579963038427843560085613, −9.715119311473587752385605383509, −9.011768506709543287781317351895, −8.619170204653682386944463403602, −8.282063256949368370755606003582, −7.64778581848165550263809601842, −7.47486373456697764431656940157, −6.78226750023478199095015630063, −6.45163948241996375246769418140, −5.75358915284655606257774059667, −5.46928895015017093649496777937, −4.55170743068002638995567224754, −4.04570128725164222587001565108, −3.25510376772767607229686246654, −3.01664450915322485633931603272, −1.97836346568506537680213813821, −1.90836237988088961761687891822, 1.90836237988088961761687891822, 1.97836346568506537680213813821, 3.01664450915322485633931603272, 3.25510376772767607229686246654, 4.04570128725164222587001565108, 4.55170743068002638995567224754, 5.46928895015017093649496777937, 5.75358915284655606257774059667, 6.45163948241996375246769418140, 6.78226750023478199095015630063, 7.47486373456697764431656940157, 7.64778581848165550263809601842, 8.282063256949368370755606003582, 8.619170204653682386944463403602, 9.011768506709543287781317351895, 9.715119311473587752385605383509, 10.04713579963038427843560085613, 10.20845153979420460309348321019, 11.11886424873715407517357760874, 11.44895139159288995580079601776

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

\(L(\frac{1}{2})\)	\(\approx\)	\(1.187138105\)
\(L(\frac12)\)	\(\approx\)	\(1.187138105\)
\(L(1)\)		not available
\(L(1)\)		not available

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