LMFDB - L-function 2-7200-40.29-c1-0-48

Dirichlet series

L(s) = 1

− 0.732i·7^-s + 2i·11^-s + 3.46·13^-s − 3.46i·17^-s − 0.535i·19^-s + 6.19i·23^-s − 6.92i·29^-s + 5.46·31^-s − 2·37^-s − 1.46·41^-s − 5.26·43^-s + 3.26i·47^-s + 6.46·49^-s + 11.4·53^-s − 7.46i·59^-s + ⋯

L(s) = 1

− 0.276i·7^-s + 0.603i·11^-s + 0.960·13^-s − 0.840i·17^-s − 0.122i·19^-s + 1.29i·23^-s − 1.28i·29^-s + 0.981·31^-s − 0.328·37^-s − 0.228·41^-s − 0.803·43^-s + 0.476i·47^-s + 0.923·49^-s + 1.57·53^-s − 0.971i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.979 + 0.200i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ 0.979 + 0.200i)$

Particular Values

$L(1)$	$\approx$	$2.084265264$
$L(\frac12)$	$\approx$	$2.084265264$
$L(\frac{3}{2})$		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 + 0.732iT - 7T^{2} $
	11	$ 1 - 2iT - 11T^{2} $
	13	$ 1 - 3.46T + 13T^{2} $
	17	$ 1 + 3.46iT - 17T^{2} $
	19	$ 1 + 0.535iT - 19T^{2} $
	23	$ 1 - 6.19iT - 23T^{2} $
	29	$ 1 + 6.92iT - 29T^{2} $
	31	$ 1 - 5.46T + 31T^{2} $
	37	$ 1 + 2T + 37T^{2} $
	41	$ 1 + 1.46T + 41T^{2} $
	43	$ 1 + 5.26T + 43T^{2} $
	47	$ 1 - 3.26iT - 47T^{2} $
	53	$ 1 - 11.4T + 53T^{2} $
	59	$ 1 + 7.46iT - 59T^{2} $
	61	$ 1 - 8.92iT - 61T^{2} $
	67	$ 1 + 10.7T + 67T^{2} $
	71	$ 1 - 5.46T + 71T^{2} $
	73	$ 1 + 7.46iT - 73T^{2} $
	79	$ 1 + 1.07T + 79T^{2} $
	83	$ 1 + 1.26T + 83T^{2} $
	89	$ 1 - 8.92T + 89T^{2} $
	97	$ 1 + 14.3iT - 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

−7.78175824333411479741029275567, −7.28620291448221916503027154258, −6.52385434199599860394205420664, −5.83161830256803883609997193149, −5.05800875851543668427791665991, −4.28865933989080338133087181301, −3.59054252411118747913977732990, −2.70279185310123182703114814394, −1.72617063808679140399505043829, −0.71234773353581599026575818348, 0.789410375204736223278888177690, 1.78852375659628581147237991058, 2.81129965797299131306630869356, 3.57865029997606879931589363229, 4.29611518287951733345620207672, 5.20958251584781267163873218890, 5.90933823841472878331716956310, 6.49246977513353498071985751252, 7.14560743085105503330310366002, 8.220533277638198936551303026750

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.084265264\)
\(L(\frac12)\)	\(\approx\)	\(2.084265264\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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