LMFDB - L-function 2-7200-8.5-c1-0-4

Dirichlet series

L(s) = 1

− 2.73·7^-s − 2i·11^-s − 3.46i·13^-s − 3.46·17^-s − 7.46i·19^-s − 4.19·23^-s + 6.92i·29^-s − 1.46·31^-s + 2i·37^-s + 5.46·41^-s − 8.73i·43^-s − 6.73·47^-s + 0.464·49^-s + 4.53i·53^-s − 0.535i·59^-s + ⋯

L(s) = 1

− 1.03·7^-s − 0.603i·11^-s − 0.960i·13^-s − 0.840·17^-s − 1.71i·19^-s − 0.874·23^-s + 1.28i·29^-s − 0.262·31^-s + 0.328i·37^-s + 0.853·41^-s − 1.33i·43^-s − 0.981·47^-s + 0.0663·49^-s + 0.623i·53^-s − 0.0697i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.258 - 0.965i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.258 - 0.965i)$

Particular Values

$L(1)$	$\approx$	$0.3344224715$
$L(\frac12)$	$\approx$	$0.3344224715$
$L(\frac{3}{2})$		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 + 2.73T + 7T^{2} $
	11	$ 1 + 2iT - 11T^{2} $
	13	$ 1 + 3.46iT - 13T^{2} $
	17	$ 1 + 3.46T + 17T^{2} $
	19	$ 1 + 7.46iT - 19T^{2} $
	23	$ 1 + 4.19T + 23T^{2} $
	29	$ 1 - 6.92iT - 29T^{2} $
	31	$ 1 + 1.46T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 - 5.46T + 41T^{2} $
	43	$ 1 + 8.73iT - 43T^{2} $
	47	$ 1 + 6.73T + 47T^{2} $
	53	$ 1 - 4.53iT - 53T^{2} $
	59	$ 1 + 0.535iT - 59T^{2} $
	61	$ 1 - 4.92iT - 61T^{2} $
	67	$ 1 - 7.26iT - 67T^{2} $
	71	$ 1 + 1.46T + 71T^{2} $
	73	$ 1 + 0.535T + 73T^{2} $
	79	$ 1 - 14.9T + 79T^{2} $
	83	$ 1 + 4.73iT - 83T^{2} $
	89	$ 1 - 4.92T + 89T^{2} $
	97	$ 1 + 6.39T + 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

−8.187381117095017400522105081721, −7.30062827633841586464286854333, −6.73552017461411541085842726468, −6.09479477217450853572123573575, −5.36573337468227404226047938845, −4.60521417854935242793493183796, −3.62648180011778115176262307797, −3.01982347968418382074957187267, −2.26240482520580481837288011335, −0.825083937110046974722486177524, 0.098979963737801417799850773461, 1.67581202925535317643983755860, 2.34091157185845732767230009187, 3.44974793824580270631494944663, 4.07902403975209386735752698549, 4.72709387336428966809989468205, 5.87929465203960158395472441480, 6.30566318486414155669849436493, 6.89014589657107799432018900659, 7.79982486377762665160155875218

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3344224715\)
\(L(\frac12)\)	\(\approx\)	\(0.3344224715\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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