LMFDB - L-function 2-4032-56.27-c1-0-51

Dirichlet series

L(s) = 1

− 3.88·5^-s + (2.62 + 0.339i)7^-s − 1.48·11^-s + 2·13^-s − 3.76i·17^-s − 0.679i·19^-s + 5.20i·23^-s + 10.0·25^-s − 7.03i·29^-s − 6.42·31^-s + (−10.1 − 1.32i)35^-s + 8.71i·37^-s + 3.16i·41^-s − 8.20·43^-s + 9.88·47^-s + ⋯

L(s) = 1

− 1.73·5^-s + (0.991 + 0.128i)7^-s − 0.446·11^-s + 0.554·13^-s − 0.913i·17^-s − 0.155i·19^-s + 1.08i·23^-s + 2.01·25^-s − 1.30i·29^-s − 1.15·31^-s + (−1.72 − 0.223i)35^-s + 1.43i·37^-s + 0.493i·41^-s − 1.25·43^-s + 1.44·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.380 + 0.924i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.380 + 0.924i)$

Particular Values

$L(1)$	$\approx$	$0.7139643403$
$L(\frac12)$	$\approx$	$0.7139643403$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-2.62 - 0.339i)T $
good	5	$ 1 + 3.88T + 5T^{2} $
	11	$ 1 + 1.48T + 11T^{2} $
	13	$ 1 - 2T + 13T^{2} $
	17	$ 1 + 3.76iT - 17T^{2} $
	19	$ 1 + 0.679iT - 19T^{2} $
	23	$ 1 - 5.20iT - 23T^{2} $
	29	$ 1 + 7.03iT - 29T^{2} $
	31	$ 1 + 6.42T + 31T^{2} $
	37	$ 1 - 8.71iT - 37T^{2} $
	41	$ 1 - 3.16iT - 41T^{2} $
	43	$ 1 + 8.20T + 43T^{2} $
	47	$ 1 - 9.88T + 47T^{2} $
	53	$ 1 - 6.42iT - 53T^{2} $
	59	$ 1 + 7.76iT - 59T^{2} $
	61	$ 1 + 12.4T + 61T^{2} $
	67	$ 1 - 8.81T + 67T^{2} $
	71	$ 1 - 12.9iT - 71T^{2} $
	73	$ 1 + 13.4iT - 73T^{2} $
	79	$ 1 + 4.44iT - 79T^{2} $
	83	$ 1 + 10.4iT - 83T^{2} $
	89	$ 1 + 10.6iT - 89T^{2} $
	97	$ 1 + 9.88iT - 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

−8.081993630367176917497905264182, −7.60776818395033584073224435893, −7.07446725541115516040152731443, −5.92807987965008552845067780536, −5.00065610878957687063820178336, −4.45286612658444236685934761393, −3.62883057054776796958481335413, −2.84278574124399048067981246442, −1.50502126150712678040164428497, −0.24594965859372605432813499733, 1.05194115337305696251236064439, 2.27953161796830063214484788464, 3.59521761066029935272673248328, 3.94811958692293021224285979296, 4.83153893068856587989572243410, 5.53722788089917803456227714201, 6.68583958516277879491271239588, 7.40627787191124486928794754007, 7.931653412889356054164953765237, 8.558297567428485891902273773447

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7139643403\)
\(L(\frac12)\)	\(\approx\)	\(0.7139643403\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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