LMFDB - L-function 2-3744-8.5-c1-0-35

Dirichlet series

L(s) = 1

− 0.218i·5^-s + 4.47·7^-s − 3.58i·11^-s + i·13^-s + 7.60·17^-s + 4.65i·19^-s − 5.21·23^-s + 4.95·25^-s + 3.39i·29^-s + 1.71·31^-s − 0.978i·35^-s + 3.06i·37^-s − 1.17·41^-s − 6.53i·43^-s + 6.69·47^-s + ⋯

L(s) = 1

− 0.0977i·5^-s + 1.69·7^-s − 1.08i·11^-s + 0.277i·13^-s + 1.84·17^-s + 1.06i·19^-s − 1.08·23^-s + 0.990·25^-s + 0.629i·29^-s + 0.307·31^-s − 0.165i·35^-s + 0.504i·37^-s − 0.183·41^-s − 0.996i·43^-s + 0.976·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0993i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.995 + 0.0993i$
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:	$(2,\ 3744,\ (\ :1/2),\ 0.995 + 0.0993i)$

Particular Values

$L(1)$	$\approx$	$2.584979390$
$L(\frac12)$	$\approx$	$2.584979390$
$L(\frac{3}{2})$		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 + 0.218iT - 5T^{2} $
	7	$ 1 - 4.47T + 7T^{2} $
	11	$ 1 + 3.58iT - 11T^{2} $
	17	$ 1 - 7.60T + 17T^{2} $
	19	$ 1 - 4.65iT - 19T^{2} $
	23	$ 1 + 5.21T + 23T^{2} $
	29	$ 1 - 3.39iT - 29T^{2} $
	31	$ 1 - 1.71T + 31T^{2} $
	37	$ 1 - 3.06iT - 37T^{2} $
	41	$ 1 + 1.17T + 41T^{2} $
	43	$ 1 + 6.53iT - 43T^{2} $
	47	$ 1 - 6.69T + 47T^{2} $
	53	$ 1 - 1.18iT - 53T^{2} $
	59	$ 1 + 9.33iT - 59T^{2} $
	61	$ 1 - 14.6iT - 61T^{2} $
	67	$ 1 - 2.10iT - 67T^{2} $
	71	$ 1 + 11.9T + 71T^{2} $
	73	$ 1 + 13.8T + 73T^{2} $
	79	$ 1 + 2.12T + 79T^{2} $
	83	$ 1 + 11.9iT - 83T^{2} $
	89	$ 1 - 10.2T + 89T^{2} $
	97	$ 1 - 16.0T + 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

−8.476666585179966149729654871223, −7.82557084490211212750044421652, −7.29742239467443001731937480268, −5.99799920712997135359194699670, −5.57067204447520478947088327077, −4.76971440577691101579350161188, −3.89388568484199370937973089753, −3.03445531496711538440059892412, −1.75274869440338282291457910911, −1.03386395813026676588732984706, 1.01350646650106426927053562596, 1.94310955903543679788559722786, 2.87759266471801207509222648409, 4.09381914090205138672210322833, 4.78480213100681525642411996666, 5.35262483856587264541431343649, 6.25215092956077090398643505143, 7.42839198281678768350190710243, 7.63450387752287554657532880591, 8.395630092237089449828535667457

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.584979390\)
\(L(\frac12)\)	\(\approx\)	\(2.584979390\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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