LMFDB - L-function 2-7200-5.4-c1-0-0

Dirichlet series

L(s) = 1

− 4.47i·7^-s + 2.23·11^-s − 4i·13^-s + 7i·17^-s − 6.70·19^-s + 4.47i·23^-s − 4.47·31^-s + 2i·37^-s − 5·41^-s − 8.94i·47^-s − 13.0·49^-s + 6i·53^-s − 8.94·59^-s + 10·61^-s + 2.23i·67^-s + ⋯

L(s) = 1

− 1.69i·7^-s + 0.674·11^-s − 1.10i·13^-s + 1.69i·17^-s − 1.53·19^-s + 0.932i·23^-s − 0.803·31^-s + 0.328i·37^-s − 0.780·41^-s − 1.30i·47^-s − 1.85·49^-s + 0.824i·53^-s − 1.16·59^-s + 1.28·61^-s + 0.273i·67^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (6049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ -0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.3628778188$
$L(\frac12)$	$\approx$	$0.3628778188$
$L(\frac{3}{2})$		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 + 4.47iT - 7T^{2} $
	11	$ 1 - 2.23T + 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 - 7iT - 17T^{2} $
	19	$ 1 + 6.70T + 19T^{2} $
	23	$ 1 - 4.47iT - 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 + 4.47T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 + 5T + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 8.94iT - 47T^{2} $
	53	$ 1 - 6iT - 53T^{2} $
	59	$ 1 + 8.94T + 59T^{2} $
	61	$ 1 - 10T + 61T^{2} $
	67	$ 1 - 2.23iT - 67T^{2} $
	71	$ 1 + 8.94T + 71T^{2} $
	73	$ 1 - 9iT - 73T^{2} $
	79	$ 1 + 4.47T + 79T^{2} $
	83	$ 1 - 11.1iT - 83T^{2} $
	89	$ 1 + 5T + 89T^{2} $
	97	$ 1 - 2iT - 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

−8.218946334767808228781761993054, −7.38329862642750107222873394945, −6.83482619873525008560630719974, −6.13322385702079983348323347613, −5.39784149886645335350533202719, −4.29865035123383011571593854429, −3.91440677537429529658819380054, −3.26124728183219900421952942671, −1.88520164164897844474793435139, −1.12130258632548584779506121241, 0.087654335613115036668467431522, 1.73422867820027683031652218276, 2.35704000147278000255294995888, 3.12631947315584158575051198782, 4.26128975382218267191807533246, 4.82164592941250853059079769600, 5.62747969889125923047843306666, 6.41068282770455354292595442098, 6.77101676489293650989197286308, 7.73142486912905488025804860605

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3628778188\)
\(L(\frac12)\)	\(\approx\)	\(0.3628778188\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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