LMFDB - L-function 2-3744-24.11-c1-0-3

Dirichlet series

L(s) = 1

− 0.327·5^-s − 2.84i·7^-s + 1.48i·11^-s − i·13^-s + 0.756i·17^-s − 4.98·19^-s − 2.71·23^-s − 4.89·25^-s + 2.16·29^-s − 3.62i·31^-s + 0.930i·35^-s + 10.0i·37^-s + 11.3i·41^-s − 2.76·43^-s + 9.45·47^-s + ⋯

L(s) = 1

− 0.146·5^-s − 1.07i·7^-s + 0.448i·11^-s − 0.277i·13^-s + 0.183i·17^-s − 1.14·19^-s − 0.566·23^-s − 0.978·25^-s + 0.402·29^-s − 0.650i·31^-s + 0.157i·35^-s + 1.65i·37^-s + 1.77i·41^-s − 0.422·43^-s + 1.37·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$-0.371 - 0.928i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (2159, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ -0.371 - 0.928i)$

Particular Values

$L(1)$	$\approx$	$0.6353324216$
$L(\frac12)$	$\approx$	$0.6353324216$
$L(\frac{3}{2})$		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 + 0.327T + 5T^{2} $
	7	$ 1 + 2.84iT - 7T^{2} $
	11	$ 1 - 1.48iT - 11T^{2} $
	17	$ 1 - 0.756iT - 17T^{2} $
	19	$ 1 + 4.98T + 19T^{2} $
	23	$ 1 + 2.71T + 23T^{2} $
	29	$ 1 - 2.16T + 29T^{2} $
	31	$ 1 + 3.62iT - 31T^{2} $
	37	$ 1 - 10.0iT - 37T^{2} $
	41	$ 1 - 11.3iT - 41T^{2} $
	43	$ 1 + 2.76T + 43T^{2} $
	47	$ 1 - 9.45T + 47T^{2} $
	53	$ 1 + 3.68T + 53T^{2} $
	59	$ 1 + 2.06iT - 59T^{2} $
	61	$ 1 - 8.28iT - 61T^{2} $
	67	$ 1 + 12.1T + 67T^{2} $
	71	$ 1 + 1.67T + 71T^{2} $
	73	$ 1 - 4.79T + 73T^{2} $
	79	$ 1 + 1.01iT - 79T^{2} $
	83	$ 1 + 12.2iT - 83T^{2} $
	89	$ 1 - 3.19iT - 89T^{2} $
	97	$ 1 + 5.62T + 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

−8.595347263938391321657812710529, −7.925213611161362165567717099035, −7.38658490541966745025401675130, −6.49279086160388001335850988022, −5.94662614297639811917418513955, −4.68950592097442374461770299337, −4.26194551541819881950663247336, −3.38554233021962467465831387397, −2.27691502246541626610032761119, −1.16937647021588890538527556877, 0.19025237178478889791683899578, 1.86648158124758708607161838077, 2.55156597904372135181691202910, 3.66483556913828885251342725554, 4.39122634803306463170311316621, 5.49436622890339845132231238998, 5.90361030604584096194450944546, 6.77224583660377731358082613720, 7.59893897214412792050809520566, 8.450802171374978651252204011421

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.6353324216\)
\(L(\frac12)\)	\(\approx\)	\(0.6353324216\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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