LMFDB - L-function 2-3744-13.12-c1-0-44

Dirichlet series

L(s) = 1

− 2.60i·7^-s + 6.60i·11^-s − 3.60·13^-s − 7.21·17^-s + 1.39i·19^-s + 5·25^-s + 7.21·29^-s − 10.6i·31^-s − 5.39i·47^-s + 0.211·49^-s + 2·53^-s − 11.8i·59^-s + 6·61^-s − 14.6i·67^-s − 15.8i·71^-s + ⋯

L(s) = 1

− 0.984i·7^-s + 1.99i·11^-s − 1.00·13^-s − 1.74·17^-s + 0.319i·19^-s + 25^-s + 1.33·29^-s − 1.90i·31^-s − 0.786i·47^-s + 0.0301·49^-s + 0.274·53^-s − 1.53i·59^-s + 0.768·61^-s − 1.78i·67^-s − 1.87i·71^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (3457, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ i)$

Particular Values

$L(1)$	$\approx$	$1.098401420$
$L(\frac12)$	$\approx$	$1.098401420$
$L(\frac{3}{2})$		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 + 3.60T $
good	5	$ 1 - 5T^{2} $
	7	$ 1 + 2.60iT - 7T^{2} $
	11	$ 1 - 6.60iT - 11T^{2} $
	17	$ 1 + 7.21T + 17T^{2} $
	19	$ 1 - 1.39iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 7.21T + 29T^{2} $
	31	$ 1 + 10.6iT - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 5.39iT - 47T^{2} $
	53	$ 1 - 2T + 53T^{2} $
	59	$ 1 + 11.8iT - 59T^{2} $
	61	$ 1 - 6T + 61T^{2} $
	67	$ 1 + 14.6iT - 67T^{2} $
	71	$ 1 + 15.8iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 3.81iT - 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

−8.188714410022765953348898517419, −7.46687973426303549897576943307, −6.91211612798683437875753461131, −6.40338724760419898388595810205, −4.93669865433510563417066547880, −4.61034845843340959257084786086, −3.90150650028817211998177541828, −2.52576631237108973117886350179, −1.88677364854305106408275258683, −0.35327816287155092958314919044, 1.04491903159100579149462532736, 2.64351768618998015962007007946, 2.83171472885535519983873340301, 4.15532604203154359013918325815, 5.05208776354983522619073681004, 5.64847975517670532648088371738, 6.55210619196176382595558219291, 7.01177430795672550093713925931, 8.297620885046749219286082070497, 8.730348733941722924493114525274

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.098401420\)
\(L(\frac12)\)	\(\approx\)	\(1.098401420\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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