LMFDB - L-function 2-2952-41.40-c1-0-1

Dirichlet series

L(s) = 1

− 2.29·5^-s + 1.04i·7^-s + 5.59i·11^-s + 2.59i·13^-s + 2.25i·17^-s − 5.17i·19^-s − 3.92·23^-s + 0.246·25^-s − 0.987i·29^-s + 10.5·31^-s − 2.39i·35^-s − 1.75·37^-s + (−5.84 + 2.62i)41^-s − 9.15·43^-s + 3.22i·47^-s + ⋯

L(s) = 1

− 1.02·5^-s + 0.394i·7^-s + 1.68i·11^-s + 0.719i·13^-s + 0.547i·17^-s − 1.18i·19^-s − 0.819·23^-s + 0.0492·25^-s − 0.183i·29^-s + 1.89·31^-s − 0.404i·35^-s − 0.288·37^-s + (−0.912 + 0.409i)41^-s − 1.39·43^-s + 0.470i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2952$ = $2^{3} \cdot 3^{2} \cdot 41$
Sign:	$-0.912 + 0.409i$
Analytic conductor:	$23.5718$
Root analytic conductor:	$4.85508$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2952} (2377, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2952,\ (\ :1/2),\ -0.912 + 0.409i)$

Particular Values

$L(1)$	$\approx$	$0.2041865640$
$L(\frac12)$	$\approx$	$0.2041865640$
$L(\frac{3}{2})$		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 $
	41	$ 1 + (5.84 - 2.62i)T $
good	5	$ 1 + 2.29T + 5T^{2} $
	7	$ 1 - 1.04iT - 7T^{2} $
	11	$ 1 - 5.59iT - 11T^{2} $
	13	$ 1 - 2.59iT - 13T^{2} $
	17	$ 1 - 2.25iT - 17T^{2} $
	19	$ 1 + 5.17iT - 19T^{2} $
	23	$ 1 + 3.92T + 23T^{2} $
	29	$ 1 + 0.987iT - 29T^{2} $
	31	$ 1 - 10.5T + 31T^{2} $
	37	$ 1 + 1.75T + 37T^{2} $
	43	$ 1 + 9.15T + 43T^{2} $
	47	$ 1 - 3.22iT - 47T^{2} $
	53	$ 1 + 6.88iT - 53T^{2} $
	59	$ 1 + 1.61T + 59T^{2} $
	61	$ 1 + 1.88T + 61T^{2} $
	67	$ 1 + 14.8iT - 67T^{2} $
	71	$ 1 + 4.92iT - 71T^{2} $
	73	$ 1 + 2.76T + 73T^{2} $
	79	$ 1 - 3.50iT - 79T^{2} $
	83	$ 1 - 4.89T + 83T^{2} $
	89	$ 1 - 2.45iT - 89T^{2} $
	97	$ 1 + 0.374iT - 97T^{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

−9.186024596959503642112760346921, −8.261461587691166479035098509576, −7.78611960199493594776592204569, −6.84606761765974607519825634317, −6.43329485390485576109441154667, −5.05166842716554899753780130358, −4.51095313007563603187811438971, −3.79135190662007772088108604961, −2.61293484277027396980944930741, −1.68043900253286977424362437977, 0.07103277161009766245274435458, 1.14751350490181379835935978652, 2.80517299811338743869106679217, 3.55904584173263795496388921866, 4.16893967262119731531712513504, 5.30222580437985839887235686374, 5.99881365750136867179254124187, 6.85636813580786540718862498853, 7.81918516862316685549625357397, 8.207590470348230142580629746806

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2041865640\)
\(L(\frac12)\)	\(\approx\)	\(0.2041865640\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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