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

Dirichlet series

L(s) = 1

− 2·2^-s + 2·4^-s − 4·7^-s − 9^-s + 8·14^-s − 4·16^-s − 4·17^-s + 2·18^-s + 8·23^-s + 6·25^-s − 8·28^-s + 4·31^-s + 8·32^-s + 8·34^-s − 2·36^-s + 4·41^-s − 16·46^-s − 24·47^-s − 2·49^-s − 12·50^-s − 8·62^-s + 4·63^-s − 8·64^-s − 8·68^-s + 24·71^-s − 12·73^-s + 20·79^-s + ⋯

L(s) = 1

− 1.41·2^-s + 4^-s − 1.51·7^-s − 1/3·9^-s + 2.13·14^-s − 16^-s − 0.970·17^-s + 0.471·18^-s + 1.66·23^-s + 6/5·25^-s − 1.51·28^-s + 0.718·31^-s + 1.41·32^-s + 1.37·34^-s − 1/3·36^-s + 0.624·41^-s − 2.35·46^-s − 3.50·47^-s − 2/7·49^-s − 1.69·50^-s − 1.01·62^-s + 0.503·63^-s − 64^-s − 0.970·68^-s + 2.84·71^-s − 1.40·73^-s + 2.25·79^-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

$L(1)$	$\approx$	$0.2239625216$
$L(\frac12)$	$\approx$	$0.2239625216$
$L(\frac{3}{2})$		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} $
	3	$C_2$	$ 1 + T^{2} $
good	5	$C_2$	$ ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) $
	7	$C_2$	$ ( 1 + 2 T + p T^{2} )^{2} $
	11	$C_2$	$ ( 1 - p T^{2} )^{2} $
	13	$C_2$	$ ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) $
	17	$C_2$	$ ( 1 + 2 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	$C_2^2$	$ 1 - 22 T^{2} + p^{2} T^{4} $
	31	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	37	$C_2^2$	$ 1 - 10 T^{2} + p^{2} T^{4} $
	41	$C_2$	$ ( 1 - 2 T + p T^{2} )^{2} $
	43	$C_2^2$	$ 1 - 70 T^{2} + p^{2} T^{4} $
	47	$C_2$	$ ( 1 + 12 T + p T^{2} )^{2} $
	53	$C_2^2$	$ 1 - 70 T^{2} + p^{2} T^{4} $
	59	$C_2^2$	$ 1 - 102 T^{2} + p^{2} T^{4} $
	61	$C_2$	$ ( 1 - p T^{2} )^{2} $
	67	$C_2^2$	$ 1 + 10 T^{2} + p^{2} T^{4} $
	71	$C_2$	$ ( 1 - 12 T + p T^{2} )^{2} $
	73	$C_2$	$ ( 1 + 6 T + p T^{2} )^{2} $
	79	$C_2$	$ ( 1 - 10 T + p T^{2} )^{2} $
	83	$C_2^2$	$ 1 + 90 T^{2} + p^{2} T^{4} $
	89	$C_2$	$ ( 1 + 10 T + p T^{2} )^{2} $
	97	$C_2$	$ ( 1 + 2 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

−18.00693396794925179402267911549, −17.66669516292985138000552330727, −16.83562473117415672804824087885, −16.49123228869151546957401212453, −15.98223877781209033394422032748, −15.27538700913625290547921376262, −14.57354599873087224304882324985, −13.50708743348838688925759206637, −13.06770714935972084475521959982, −12.43052393866941433087292838450, −11.14776256340104007773522735073, −11.00575814520349641732290778350, −9.848579168978871689408502955344, −9.529905838361445048625055372990, −8.763614598696003267997171626681, −8.076922677812066366448595950498, −6.73600798783002870211285003472, −6.63334300779775235394940480185, −4.87562501130936063565193053851, −3.03882077536188858522288571892, 3.03882077536188858522288571892, 4.87562501130936063565193053851, 6.63334300779775235394940480185, 6.73600798783002870211285003472, 8.076922677812066366448595950498, 8.763614598696003267997171626681, 9.529905838361445048625055372990, 9.848579168978871689408502955344, 11.00575814520349641732290778350, 11.14776256340104007773522735073, 12.43052393866941433087292838450, 13.06770714935972084475521959982, 13.50708743348838688925759206637, 14.57354599873087224304882324985, 15.27538700913625290547921376262, 15.98223877781209033394422032748, 16.49123228869151546957401212453, 16.83562473117415672804824087885, 17.66669516292985138000552330727, 18.00693396794925179402267911549

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2239625216\)
\(L(\frac12)\)	\(\approx\)	\(0.2239625216\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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