LMFDB - L-function 4-3120e2-1.1-c0e2-0-1

Dirichlet series

L(s) = 1

− 2·2^-s + 3·4^-s − 4·8^-s − 9^-s + 2·11^-s − 2·13^-s + 5·16^-s + 2·18^-s − 4·22^-s − 25^-s + 4·26^-s − 6·32^-s − 3·36^-s + 4·43^-s + 6·44^-s − 2·47^-s + 2·50^-s − 6·52^-s − 2·59^-s + 2·61^-s + 7·64^-s + 4·72^-s + 81^-s − 8·86^-s − 8·88^-s + 4·94^-s − 2·99^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$9734400$ = $2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$
Sign:	$1$
Analytic conductor:	$2.42450$
Root analytic conductor:	$1.24783$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 9734400,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4158846090$
$L(\frac12)$	$\approx$	$0.4158846090$
$L(1)$		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	2	$C_1$	$ ( 1 + T )^{2} $
	3	$C_2$	$ 1 + T^{2} $
	5	$C_2$	$ 1 + T^{2} $
	13	$C_1$	$ ( 1 + T )^{2} $
good	7	$C_2^2$	$ 1 + T^{4} $
	11	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T^{2} ) $
	17	$C_2^2$	$ 1 + T^{4} $
	19	$C_2^2$	$ 1 + T^{4} $
	23	$C_2^2$	$ 1 + T^{4} $
	29	$C_2^2$	$ 1 + T^{4} $
	31	$C_2$	$ ( 1 + T^{2} )^{2} $
	37	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	41	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	43	$C_1$	$ ( 1 - T )^{4} $
	47	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	53	$C_2$	$ ( 1 + T^{2} )^{2} $
	59	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	61	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T^{2} ) $
	67	$C_1$$\times$$C_1$	$ ( 1 - T )^{2}( 1 + T )^{2} $
	71	$C_2$	$ ( 1 + T^{2} )^{2} $
	73	$C_2^2$	$ 1 + T^{4} $
	79	$C_2$	$ ( 1 + T^{2} )^{2} $
	83	$C_2$	$ ( 1 + T^{2} )^{2} $
	89	$C_2$	$ ( 1 + T^{2} )^{2} $
	97	$C_2^2$	$ 1 + 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

−9.125741576254795658406416886821, −8.799405439354017893493385982619, −8.381068057278989741089696093219, −7.958106368312314234452791060958, −7.53440973036262623961836436184, −7.50476470485201116087922700329, −6.90241159447246556914437943356, −6.47172921104496432294442512582, −6.40740587225115393998658764695, −5.73137201002609853628825669864, −5.57260196876373121860959732616, −4.97681882775423615675895223979, −4.15686363831949430669566830705, −3.99313671581177279166493107061, −3.15773129903802495565451824828, −2.89180057647746774688175045844, −2.31035081054943965630347052602, −1.96699437227849585332434274271, −1.32429914204231490987548056718, −0.56452836210169933209185644603, 0.56452836210169933209185644603, 1.32429914204231490987548056718, 1.96699437227849585332434274271, 2.31035081054943965630347052602, 2.89180057647746774688175045844, 3.15773129903802495565451824828, 3.99313671581177279166493107061, 4.15686363831949430669566830705, 4.97681882775423615675895223979, 5.57260196876373121860959732616, 5.73137201002609853628825669864, 6.40740587225115393998658764695, 6.47172921104496432294442512582, 6.90241159447246556914437943356, 7.50476470485201116087922700329, 7.53440973036262623961836436184, 7.958106368312314234452791060958, 8.381068057278989741089696093219, 8.799405439354017893493385982619, 9.125741576254795658406416886821

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.4158846090\)
\(L(\frac12)\)	\(\approx\)	\(0.4158846090\)
\(L(1)\)		not available
\(L(1)\)		not available

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