LMFDB - L-function 2-252-21.20-c7-0-3

Dirichlet series

L(s) = 1

+ 438.·5^-s + (−907. − 20.5i)7^-s − 7.41e3i·11^-s + 2.70e3i·13^-s − 3.34e4·17^-s + 4.84e4i·19^-s − 4.66e4i·23^-s + 1.14e5·25^-s + 1.59e5i·29^-s + 2.14e5i·31^-s + (−3.97e5 − 9.02e3i)35^-s − 3.94e5·37^-s + 1.57e5·41^-s − 6.53e5·43^-s + 9.95e5·47^-s + ⋯

L(s) = 1

+ 1.56·5^-s + (−0.999 − 0.0226i)7^-s − 1.67i·11^-s + 0.341i·13^-s − 1.65·17^-s + 1.62i·19^-s − 0.799i·23^-s + 1.45·25^-s + 1.21i·29^-s + 1.29i·31^-s + (−1.56 − 0.0355i)35^-s − 1.27·37^-s + 0.357·41^-s − 1.25·43^-s + 1.39·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$252$ = $2^{2} \cdot 3^{2} \cdot 7$
Sign:	$-0.558 - 0.829i$
Analytic conductor:	$78.7210$
Root analytic conductor:	$8.87248$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{252} (125, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 252,\ (\ :7/2),\ -0.558 - 0.829i)$

Particular Values

$L(4)$	$\approx$	$0.9608030988$
$L(\frac12)$	$\approx$	$0.9608030988$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + (907. + 20.5i)T $
good	5	$ 1 - 438.T + 7.81e4T^{2} $
	11	$ 1 + 7.41e3iT - 1.94e7T^{2} $
	13	$ 1 - 2.70e3iT - 6.27e7T^{2} $
	17	$ 1 + 3.34e4T + 4.10e8T^{2} $
	19	$ 1 - 4.84e4iT - 8.93e8T^{2} $
	23	$ 1 + 4.66e4iT - 3.40e9T^{2} $
	29	$ 1 - 1.59e5iT - 1.72e10T^{2} $
	31	$ 1 - 2.14e5iT - 2.75e10T^{2} $
	37	$ 1 + 3.94e5T + 9.49e10T^{2} $
	41	$ 1 - 1.57e5T + 1.94e11T^{2} $
	43	$ 1 + 6.53e5T + 2.71e11T^{2} $
	47	$ 1 - 9.95e5T + 5.06e11T^{2} $
	53	$ 1 - 9.18e5iT - 1.17e12T^{2} $
	59	$ 1 - 4.26e5T + 2.48e12T^{2} $
	61	$ 1 - 4.09e5iT - 3.14e12T^{2} $
	67	$ 1 - 6.18e5T + 6.06e12T^{2} $
	71	$ 1 + 3.96e6iT - 9.09e12T^{2} $
	73	$ 1 - 3.84e6iT - 1.10e13T^{2} $
	79	$ 1 + 1.41e6T + 1.92e13T^{2} $
	83	$ 1 - 3.32e6T + 2.71e13T^{2} $
	89	$ 1 + 1.18e7T + 4.42e13T^{2} $
	97	$ 1 - 1.26e7iT - 8.07e13T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.75649904581888820210865354631, −10.29590912870465145054754325396, −9.085129835687879166528814431339, −8.637232692106656838919730366963, −6.75051752132145359754431135235, −6.19580213740590127192137699882, −5.30987707100000370851901280260, −3.63887114526902902351473528636, −2.53259645265121289728780573518, −1.31611467938779697984593686586, 0.20154120910808619819896072550, 1.94212912626462876108530861554, 2.59298238727386906038897372949, 4.33655428389677930146089291116, 5.44893592809211345515949391257, 6.53081491178769414621409027345, 7.14758627147662990054967178536, 8.913315346341781743805726879095, 9.630629580802005690957911621525, 10.11683207545435088339161141081

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.9608030988\)
\(L(\frac12)\)	\(\approx\)	\(0.9608030988\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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