LMFDB - L-function 2-2736-76.75-c1-0-30

Dirichlet series

L(s) = 1

− 2.37·5^-s + 2.52i·7^-s − 2.52i·11^-s + 1.58i·13^-s + 0.372·17^-s + (−4 + 1.73i)19^-s − 1.87i·23^-s + 0.627·25^-s + 3.16i·29^-s + 2.74·31^-s − 5.98i·35^-s + 1.58i·37^-s − 6.92i·41^-s − 0.644i·43^-s − 0.939i·47^-s + ⋯

L(s) = 1

− 1.06·5^-s + 0.954i·7^-s − 0.761i·11^-s + 0.439i·13^-s + 0.0902·17^-s + (−0.917 + 0.397i)19^-s − 0.391i·23^-s + 0.125·25^-s + 0.588i·29^-s + 0.492·31^-s − 1.01i·35^-s + 0.260i·37^-s − 1.08i·41^-s − 0.0983i·43^-s − 0.137i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.114 + 0.993i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ -0.114 + 0.993i)$

Particular Values

$L(1)$	$\approx$	$0.6285900663$
$L(\frac12)$	$\approx$	$0.6285900663$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (4 - 1.73i)T $
good	5	$ 1 + 2.37T + 5T^{2} $
	7	$ 1 - 2.52iT - 7T^{2} $
	11	$ 1 + 2.52iT - 11T^{2} $
	13	$ 1 - 1.58iT - 13T^{2} $
	17	$ 1 - 0.372T + 17T^{2} $
	23	$ 1 + 1.87iT - 23T^{2} $
	29	$ 1 - 3.16iT - 29T^{2} $
	31	$ 1 - 2.74T + 31T^{2} $
	37	$ 1 - 1.58iT - 37T^{2} $
	41	$ 1 + 6.92iT - 41T^{2} $
	43	$ 1 + 0.644iT - 43T^{2} $
	47	$ 1 + 0.939iT - 47T^{2} $
	53	$ 1 + 10.0iT - 53T^{2} $
	59	$ 1 + 4T + 59T^{2} $
	61	$ 1 + 0.372T + 61T^{2} $
	67	$ 1 + 13.4T + 67T^{2} $
	71	$ 1 + 4T + 71T^{2} $
	73	$ 1 - 13.1T + 73T^{2} $
	79	$ 1 - 6.74T + 79T^{2} $
	83	$ 1 + 3.46iT - 83T^{2} $
	89	$ 1 + 13.2iT - 89T^{2} $
	97	$ 1 + 13.2iT - 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

−8.567035649293616778110143142171, −8.053641848725363207330039010344, −7.13416005820723001972481556817, −6.30545369879567739496631235979, −5.57910366018975732276592685003, −4.60683008674190201247236088569, −3.80203283844520693579824779265, −2.96247962349701155859142772982, −1.86533902966586833474346644876, −0.23652182474508181986311652935, 1.04070646736043895234988613409, 2.46486827481052905037631686302, 3.58944344006895952516585196945, 4.26170588316308541084313767528, 4.85944625468801963236117433567, 6.09805229614701810385523546310, 6.89035487589820193302034558686, 7.70770103100857121984386685090, 7.936691265399016940185218364399, 9.003638457560240625581793034891

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6285900663\)
\(L(\frac12)\)	\(\approx\)	\(0.6285900663\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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