LMFDB - L-function 2-4032-8.5-c1-0-4

Dirichlet series

L(s) = 1

− 0.732i·5^-s + 7^-s + 2.73i·11^-s − 4i·13^-s − 4.19·17^-s + 3.46i·19^-s − 4.73·23^-s + 4.46·25^-s + 7.46i·29^-s − 2·31^-s − 0.732i·35^-s − 2i·37^-s − 9.66·41^-s + 8.92i·43^-s + 4·47^-s + ⋯

L(s) = 1

− 0.327i·5^-s + 0.377·7^-s + 0.823i·11^-s − 1.10i·13^-s − 1.01·17^-s + 0.794i·19^-s − 0.986·23^-s + 0.892·25^-s + 1.38i·29^-s − 0.359·31^-s − 0.123i·35^-s − 0.328i·37^-s − 1.50·41^-s + 1.36i·43^-s + 0.583·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.258 - 0.965i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (2017, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.258 - 0.965i)$

Particular Values

$L(1)$	$\approx$	$1.044817358$
$L(\frac12)$	$\approx$	$1.044817358$
$L(\frac{3}{2})$		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 - T $
good	5	$ 1 + 0.732iT - 5T^{2} $
	11	$ 1 - 2.73iT - 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 + 4.19T + 17T^{2} $
	19	$ 1 - 3.46iT - 19T^{2} $
	23	$ 1 + 4.73T + 23T^{2} $
	29	$ 1 - 7.46iT - 29T^{2} $
	31	$ 1 + 2T + 31T^{2} $
	37	$ 1 + 2iT - 37T^{2} $
	41	$ 1 + 9.66T + 41T^{2} $
	43	$ 1 - 8.92iT - 43T^{2} $
	47	$ 1 - 4T + 47T^{2} $
	53	$ 1 - 3.46iT - 53T^{2} $
	59	$ 1 + 8iT - 59T^{2} $
	61	$ 1 - 8.39iT - 61T^{2} $
	67	$ 1 - 6.39iT - 67T^{2} $
	71	$ 1 - 3.66T + 71T^{2} $
	73	$ 1 + 8.53T + 73T^{2} $
	79	$ 1 - 12.3T + 79T^{2} $
	83	$ 1 - 2.53iT - 83T^{2} $
	89	$ 1 + 12.1T + 89T^{2} $
	97	$ 1 + 2.39T + 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

−8.580563095062117273103364618972, −7.990036483655785879006957279459, −7.24529442083932583643244180098, −6.50056816101434545993521791656, −5.56756806259588147343239847782, −4.93518315202610551654211155459, −4.18787883299432359224542559178, −3.23067072893584540880053657707, −2.18219372037005450272230572390, −1.24405212232869413154634870409, 0.29753315247689107408285330784, 1.79578379662788539949873156126, 2.57368479375650189222143078429, 3.66741921796943744819764285675, 4.40450300818983863673685508024, 5.18682411466633153839234183142, 6.17933426658338521627382551010, 6.70992808127410050873996569173, 7.42066527055007788782498139652, 8.406600224708347853609186791363

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.044817358\)
\(L(\frac12)\)	\(\approx\)	\(1.044817358\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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