LMFDB - L-function 4-24e2-1.1-c1e2-0-1

Dirichlet series

L(s) = 1

− 2·3^-s − 2·4^-s + 9^-s + 4·12^-s + 4·16^-s + 4·19^-s − 10·25^-s + 4·27^-s − 2·36^-s − 20·43^-s − 8·48^-s + 14·49^-s − 8·57^-s − 8·64^-s + 28·67^-s + 4·73^-s + 20·75^-s − 8·76^-s − 11·81^-s − 20·97^-s + 20·100^-s − 8·108^-s + 14·121^-s + 127^-s + 40·129^-s + 131^-s + 137^-s + ⋯

L(s) = 1

− 1.15·3^-s − 4^-s + 1/3·9^-s + 1.15·12^-s + 16^-s + 0.917·19^-s − 2·25^-s + 0.769·27^-s − 1/3·36^-s − 3.04·43^-s − 1.15·48^-s + 2·49^-s − 1.05·57^-s − 64^-s + 3.42·67^-s + 0.468·73^-s + 2.30·75^-s − 0.917·76^-s − 1.22·81^-s − 2.03·97^-s + 2·100^-s − 0.769·108^-s + 1.27·121^-s + 0.0887·127^-s + 3.52·129^-s + 0.0873·131^-s + 0.0854·137^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$1$
Analytic conductor:	$0.0367262$
Root analytic conductor:	$0.437768$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 576,\ (\ :1/2, 1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$0.2878147484$
$L(\frac12)$	$\approx$	$0.2878147484$
$L(\frac{3}{2})$		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^{2} $
	3	$C_2$	$ 1 + 2 T + p T^{2} $
good	5	$C_2$	$ ( 1 + p T^{2} )^{2} $
	7	$C_2$	$ ( 1 - p T^{2} )^{2} $
	11	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	13	$C_2$	$ ( 1 - p T^{2} )^{2} $
	17	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	19	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	23	$C_2$	$ ( 1 + p T^{2} )^{2} $
	29	$C_2$	$ ( 1 + p T^{2} )^{2} $
	31	$C_2$	$ ( 1 - p T^{2} )^{2} $
	37	$C_2$	$ ( 1 - p T^{2} )^{2} $
	41	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	43	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
	47	$C_2$	$ ( 1 + p T^{2} )^{2} $
	53	$C_2$	$ ( 1 + p T^{2} )^{2} $
	59	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	61	$C_2$	$ ( 1 - p T^{2} )^{2} $
	67	$C_2$	$ ( 1 - 14 T + p T^{2} )^{2} $
	71	$C_2$	$ ( 1 + p T^{2} )^{2} $
	73	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	79	$C_2$	$ ( 1 - p T^{2} )^{2} $
	83	$C_2$	$ ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) $
	89	$C_2$	$ ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) $
	97	$C_2$	$ ( 1 + 10 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

−17.86271258287136632536477627055, −17.71275726124330819883970306751, −16.80125482013019331873847309096, −16.70501923385416248717129689884, −15.64610061145699713347539417384, −15.19730047189340621458262065414, −14.15078414747634405407289801866, −13.75877200386722474683296855826, −13.07485440210465344807068665664, −12.22728533480882669071683978630, −11.75089254705110530455108571385, −11.08980142422841485939985229545, −10.05403923244183463898302503844, −9.720994441844088178423615033944, −8.628873380779201877048120215204, −7.86668938059784971565526833646, −6.71952475783811132454670618982, −5.65243967610745928202997489765, −5.09127124120506387813062870719, −3.79734348170821547926452829026, 3.79734348170821547926452829026, 5.09127124120506387813062870719, 5.65243967610745928202997489765, 6.71952475783811132454670618982, 7.86668938059784971565526833646, 8.628873380779201877048120215204, 9.720994441844088178423615033944, 10.05403923244183463898302503844, 11.08980142422841485939985229545, 11.75089254705110530455108571385, 12.22728533480882669071683978630, 13.07485440210465344807068665664, 13.75877200386722474683296855826, 14.15078414747634405407289801866, 15.19730047189340621458262065414, 15.64610061145699713347539417384, 16.70501923385416248717129689884, 16.80125482013019331873847309096, 17.71275726124330819883970306751, 17.86271258287136632536477627055

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2878147484\)
\(L(\frac12)\)	\(\approx\)	\(0.2878147484\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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