LMFDB - L-function 2-2400-24.11-c1-0-10

Dirichlet series

L(s) = 1

− 1.73i·3^-s − 2.99·9^-s + 6.92i·17^-s + 4·19^-s − 8.94·23^-s + 5.19i·27^-s + 7.74i·31^-s − 8.94·47^-s + 7·49^-s + 11.9·51^-s + 4.47·53^-s − 6.92i·57^-s + 15.4i·61^-s + 15.4i·69^-s + 7.74i·79^-s + ⋯

L(s) = 1

− 0.999i·3^-s − 0.999·9^-s + 1.68i·17^-s + 0.917·19^-s − 1.86·23^-s + 0.999i·27^-s + 1.39i·31^-s − 1.30·47^-s + 49^-s + 1.68·51^-s + 0.614·53^-s − 0.917i·57^-s + 1.98i·61^-s + 1.86i·69^-s + 0.871i·79^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$0.612 - 0.790i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (2351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ 0.612 - 0.790i)$

Particular Values

$L(1)$	$\approx$	$1.091870824$
$L(\frac12)$	$\approx$	$1.091870824$
$L(\frac{3}{2})$		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 + 1.73iT $
	5	$ 1 $
good	7	$ 1 - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 6.92iT - 17T^{2} $
	19	$ 1 - 4T + 19T^{2} $
	23	$ 1 + 8.94T + 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 - 7.74iT - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 8.94T + 47T^{2} $
	53	$ 1 - 4.47T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 15.4iT - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 73T^{2} $
	79	$ 1 - 7.74iT - 79T^{2} $
	83	$ 1 - 3.46iT - 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 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

−8.770028752031780814953370557471, −8.276667262366009347868217420889, −7.56896994254133447837874540263, −6.77683714649729463163547048774, −6.00851601602646929213618165406, −5.42784428290185321779913141118, −4.15150737967276833822002584368, −3.25237802149363320592251824163, −2.11806964897902627891028492199, −1.24886155361514589552201917328, 0.37958263831386666004294485887, 2.20947453172413770528506591632, 3.17528537114985370554770379499, 4.02803521959377076695882249305, 4.86789457165996237329452454420, 5.57188175236577507502216735272, 6.37492829890757152923690986950, 7.48783039432950710879151652064, 8.092152390571993925684056467206, 9.065553500223775562541418370866

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.091870824\)
\(L(\frac12)\)	\(\approx\)	\(1.091870824\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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