LMFDB - L-function 2-7200-8.5-c1-0-21

Dirichlet series

L(s) = 1

− 2.64·7^-s − 1.51i·11^-s + 3.87i·13^-s − 3.31·17^-s − 7.08i·19^-s + 4.82·23^-s + 2.18i·29^-s + 7.36·31^-s + 7.87i·37^-s − 8.72·41^-s − 1.01i·43^-s − 7.08·47^-s − 0.0164·49^-s − 4.50i·53^-s − 6.79i·59^-s + ⋯

L(s) = 1

− 0.998·7^-s − 0.456i·11^-s + 1.07i·13^-s − 0.803·17^-s − 1.62i·19^-s + 1.00·23^-s + 0.405i·29^-s + 1.32·31^-s + 1.29i·37^-s − 1.36·41^-s − 0.155i·43^-s − 1.03·47^-s − 0.00234·49^-s − 0.619i·53^-s − 0.885i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.318 - 0.947i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (3601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ 0.318 - 0.947i)$

Particular Values

$L(1)$	$\approx$	$1.106726406$
$L(\frac12)$	$\approx$	$1.106726406$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 2.64T + 7T^{2} $
	11	$ 1 + 1.51iT - 11T^{2} $
	13	$ 1 - 3.87iT - 13T^{2} $
	17	$ 1 + 3.31T + 17T^{2} $
	19	$ 1 + 7.08iT - 19T^{2} $
	23	$ 1 - 4.82T + 23T^{2} $
	29	$ 1 - 2.18iT - 29T^{2} $
	31	$ 1 - 7.36T + 31T^{2} $
	37	$ 1 - 7.87iT - 37T^{2} $
	41	$ 1 + 8.72T + 41T^{2} $
	43	$ 1 + 1.01iT - 43T^{2} $
	47	$ 1 + 7.08T + 47T^{2} $
	53	$ 1 + 4.50iT - 53T^{2} $
	59	$ 1 + 6.79iT - 59T^{2} $
	61	$ 1 + 3.60iT - 61T^{2} $
	67	$ 1 + 1.01iT - 67T^{2} $
	71	$ 1 + 6.72T + 71T^{2} $
	73	$ 1 - 15.5T + 73T^{2} $
	79	$ 1 - 7.36T + 79T^{2} $
	83	$ 1 - 7.74iT - 83T^{2} $
	89	$ 1 + 14.7T + 89T^{2} $
	97	$ 1 - 11.1T + 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

−8.252117278171235004670260365516, −7.03241535190055387894217114057, −6.66529054069002692500451948255, −6.30565839103044377211435897864, −4.97016666853559235132451290862, −4.72348336340524920484771760104, −3.54783456215211583105874326736, −2.98294124446256324467831359992, −2.08588431596511163593280382256, −0.818677890534892395239808666694, 0.34288606396427648747733880778, 1.59379318300341490358402332677, 2.68770500252846364712183001335, 3.32794864041759815134838386909, 4.12549418135309122802404180687, 4.98099433437127292603693249595, 5.79108022029402463209342235939, 6.36899769184134688156363273554, 7.04697737873044758835092420201, 7.79453628483475542995105581318

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.106726406\)
\(L(\frac12)\)	\(\approx\)	\(1.106726406\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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