LMFDB - L-function 2-3744-8.5-c1-0-55

Dirichlet series

L(s) = 1

− 3.46i·5^-s + 1.26·7^-s − 4.73i·11^-s + i·13^-s − 5.46·17^-s − 0.732i·19^-s + 4·23^-s − 6.99·25^-s − 2i·29^-s + 6.73·31^-s − 4.39i·35^-s − 8.92i·37^-s − 8.92·41^-s + 0.535i·43^-s + 6.73·47^-s + ⋯

L(s) = 1

− 1.54i·5^-s + 0.479·7^-s − 1.42i·11^-s + 0.277i·13^-s − 1.32·17^-s − 0.167i·19^-s + 0.834·23^-s − 1.39·25^-s − 0.371i·29^-s + 1.20·31^-s − 0.742i·35^-s − 1.46i·37^-s − 1.39·41^-s + 0.0817i·43^-s + 0.981·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$-0.965 + 0.258i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (1873, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ -0.965 + 0.258i)$

Particular Values

$L(1)$	$\approx$	$1.336829911$
$L(\frac12)$	$\approx$	$1.336829911$
$L(\frac{3}{2})$		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 $
	13	$ 1 - iT $
good	5	$ 1 + 3.46iT - 5T^{2} $
	7	$ 1 - 1.26T + 7T^{2} $
	11	$ 1 + 4.73iT - 11T^{2} $
	17	$ 1 + 5.46T + 17T^{2} $
	19	$ 1 + 0.732iT - 19T^{2} $
	23	$ 1 - 4T + 23T^{2} $
	29	$ 1 + 2iT - 29T^{2} $
	31	$ 1 - 6.73T + 31T^{2} $
	37	$ 1 + 8.92iT - 37T^{2} $
	41	$ 1 + 8.92T + 41T^{2} $
	43	$ 1 - 0.535iT - 43T^{2} $
	47	$ 1 - 6.73T + 47T^{2} $
	53	$ 1 - 2.92iT - 53T^{2} $
	59	$ 1 + 10.1iT - 59T^{2} $
	61	$ 1 - 2.92iT - 61T^{2} $
	67	$ 1 + 0.732iT - 67T^{2} $
	71	$ 1 + 8.19T + 71T^{2} $
	73	$ 1 + 7.46T + 73T^{2} $
	79	$ 1 + 5.46T + 79T^{2} $
	83	$ 1 - 3.26iT - 83T^{2} $
	89	$ 1 - 17.3T + 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

−8.420884619767691215392730505391, −7.63963710685794925885870181253, −6.58834216968004274248243753508, −5.85202721364985541139427874737, −5.01733897881346374666627078596, −4.53652098156386998574919146442, −3.63274396101479278127290982258, −2.42705658725803742897242150660, −1.31508394645461363202472034148, −0.39414287635747183105339635758, 1.62958995975759591194702501662, 2.52948044814699542461169622185, 3.21904758878717879894230679270, 4.37510753131415477331869164057, 4.91620738448860821009956149025, 6.08985061025075853243178585576, 6.88275490615450642198634731343, 7.10344488962249749750879296522, 8.015260763211827948140487074752, 8.796113526778944632928854321111

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.336829911\)
\(L(\frac12)\)	\(\approx\)	\(1.336829911\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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