LMFDB - L-function 2-3744-24.11-c1-0-29

Dirichlet series

L(s) = 1

− 0.802·5^-s + 1.93i·7^-s + 1.56i·11^-s − i·13^-s − 3.54i·17^-s − 4.68·19^-s + 8.86·23^-s − 4.35·25^-s − 8.14·29^-s − 10.3i·31^-s − 1.55i·35^-s − 7.05i·37^-s + 4.31i·41^-s + 1.21·43^-s + 10.6·47^-s + ⋯

L(s) = 1

− 0.359·5^-s + 0.732i·7^-s + 0.471i·11^-s − 0.277i·13^-s − 0.860i·17^-s − 1.07·19^-s + 1.84·23^-s − 0.871·25^-s − 1.51·29^-s − 1.86i·31^-s − 0.263i·35^-s − 1.15i·37^-s + 0.673i·41^-s + 0.184·43^-s + 1.55·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.656 + 0.754i$
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:	$(2,\ 3744,\ (\ :1/2),\ 0.656 + 0.754i)$

Particular Values

$L(1)$	$\approx$	$1.336135378$
$L(\frac12)$	$\approx$	$1.336135378$
$L(\frac{3}{2})$		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 + 0.802T + 5T^{2} $
	7	$ 1 - 1.93iT - 7T^{2} $
	11	$ 1 - 1.56iT - 11T^{2} $
	17	$ 1 + 3.54iT - 17T^{2} $
	19	$ 1 + 4.68T + 19T^{2} $
	23	$ 1 - 8.86T + 23T^{2} $
	29	$ 1 + 8.14T + 29T^{2} $
	31	$ 1 + 10.3iT - 31T^{2} $
	37	$ 1 + 7.05iT - 37T^{2} $
	41	$ 1 - 4.31iT - 41T^{2} $
	43	$ 1 - 1.21T + 43T^{2} $
	47	$ 1 - 10.6T + 47T^{2} $
	53	$ 1 - 1.19T + 53T^{2} $
	59	$ 1 - 8.98iT - 59T^{2} $
	61	$ 1 + 2.66iT - 61T^{2} $
	67	$ 1 - 9.46T + 67T^{2} $
	71	$ 1 - 2.27T + 71T^{2} $
	73	$ 1 + 12.1T + 73T^{2} $
	79	$ 1 + 0.0205iT - 79T^{2} $
	83	$ 1 - 0.956iT - 83T^{2} $
	89	$ 1 + 4.61iT - 89T^{2} $
	97	$ 1 - 15.1T + 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

−8.500030640279321739401654124467, −7.46132185246914058693368803824, −7.23739924729832688782739916722, −6.01027953509492203725043768020, −5.53138951526445303081797119349, −4.55416707033128844256967673911, −3.84701986040649800517145446628, −2.72567686773483815239753900416, −2.03515266274660908863235316280, −0.47513689364554808423080220251, 0.945187964969804317165551230490, 2.06759080467796518883633533624, 3.33810111852727070219672171213, 3.90104061224792899314158040029, 4.75719411318335722642017497103, 5.62278819230343007096666979025, 6.52718376951253014553165395311, 7.13816077682316657881896520772, 7.82908168147149785379719385023, 8.721263292927453412772372898308

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

\(L(1)\)	\(\approx\)	\(1.336135378\)
\(L(\frac12)\)	\(\approx\)	\(1.336135378\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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