LMFDB - L-function 2-7200-40.29-c1-0-76

Dirichlet series

L(s) = 1

− 3.62i·7^-s + 6.20i·11^-s + 0.578·13^-s + 1.42i·17^-s − 5.62i·19^-s − 5.62i·23^-s + 2i·29^-s + 2.57·31^-s − 7.83·37^-s − 5.25·41^-s − 7.25·43^-s − 6.78i·47^-s − 6.15·49^-s − 2·53^-s + 2.20i·59^-s + ⋯

L(s) = 1

− 1.37i·7^-s + 1.87i·11^-s + 0.160·13^-s + 0.344i·17^-s − 1.29i·19^-s − 1.17i·23^-s + 0.371i·29^-s + 0.463·31^-s − 1.28·37^-s − 0.820·41^-s − 1.10·43^-s − 0.989i·47^-s − 0.879·49^-s − 0.274·53^-s + 0.287i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.935 + 0.353i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.935 + 0.353i)$

Particular Values

$L(1)$	$\approx$	$0.6197570241$
$L(\frac12)$	$\approx$	$0.6197570241$
$L(\frac{3}{2})$		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 + 3.62iT - 7T^{2} $
	11	$ 1 - 6.20iT - 11T^{2} $
	13	$ 1 - 0.578T + 13T^{2} $
	17	$ 1 - 1.42iT - 17T^{2} $
	19	$ 1 + 5.62iT - 19T^{2} $
	23	$ 1 + 5.62iT - 23T^{2} $
	29	$ 1 - 2iT - 29T^{2} $
	31	$ 1 - 2.57T + 31T^{2} $
	37	$ 1 + 7.83T + 37T^{2} $
	41	$ 1 + 5.25T + 41T^{2} $
	43	$ 1 + 7.25T + 43T^{2} $
	47	$ 1 + 6.78iT - 47T^{2} $
	53	$ 1 + 2T + 53T^{2} $
	59	$ 1 - 2.20iT - 59T^{2} $
	61	$ 1 - 12.4iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 - 8.41T + 71T^{2} $
	73	$ 1 + 6iT - 73T^{2} $
	79	$ 1 - 5.42T + 79T^{2} $
	83	$ 1 + 3.25T + 83T^{2} $
	89	$ 1 + 13.2T + 89T^{2} $
	97	$ 1 + 4.84iT - 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

−7.41530353556435775781498462862, −6.86670585820546378462016185316, −6.65192066733545819092463466631, −5.29591845412098328858669024270, −4.61511835500759181979565076512, −4.20249181071985733387717906263, −3.26271011873986072133740578230, −2.22543939261080394598991939710, −1.36345549334108916703780607496, −0.14945877585991323928488059560, 1.27848470901130704856010519012, 2.23232521015764167104602020700, 3.30434479186679446802700638066, 3.53133240976739450476386198701, 4.91482306168587119292789423671, 5.56659372080511354608147950427, 5.99616944033927139771601751754, 6.64326008806742989802986705564, 7.75316406009936403658338519217, 8.365204948946388388635209902862

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6197570241\)
\(L(\frac12)\)	\(\approx\)	\(0.6197570241\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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