LMFDB - L-function 2-3042-13.12-c1-0-18

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 3.73i·5^-s + 2.73i·7^-s + i·8^-s − 3.73·10^-s + 1.26i·11^-s + 2.73·14^-s + 16^-s − 5.73·17^-s + 4.73i·19^-s + 3.73i·20^-s + 1.26·22^-s + 4.19·23^-s − 8.92·25^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 1.66i·5^-s + 1.03i·7^-s + 0.353i·8^-s − 1.18·10^-s + 0.382i·11^-s + 0.730·14^-s + 0.250·16^-s − 1.39·17^-s + 1.08i·19^-s + 0.834i·20^-s + 0.270·22^-s + 0.874·23^-s − 1.78·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3042$ = $2 \cdot 3^{2} \cdot 13^{2}$
Sign:	$0.960 + 0.277i$
Analytic conductor:	$24.2904$
Root analytic conductor:	$4.92853$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3042} (1351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3042,\ (\ :1/2),\ 0.960 + 0.277i)$

Particular Values

$L(1)$	$\approx$	$1.387873465$
$L(\frac12)$	$\approx$	$1.387873465$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	13	$ 1 $
good	5	$ 1 + 3.73iT - 5T^{2} $
	7	$ 1 - 2.73iT - 7T^{2} $
	11	$ 1 - 1.26iT - 11T^{2} $
	17	$ 1 + 5.73T + 17T^{2} $
	19	$ 1 - 4.73iT - 19T^{2} $
	23	$ 1 - 4.19T + 23T^{2} $
	29	$ 1 - 4.46T + 29T^{2} $
	31	$ 1 - 1.46iT - 31T^{2} $
	37	$ 1 + 3.53iT - 37T^{2} $
	41	$ 1 - 9.39iT - 41T^{2} $
	43	$ 1 - 9.66T + 43T^{2} $
	47	$ 1 - 2.19iT - 47T^{2} $
	53	$ 1 - 6.46T + 53T^{2} $
	59	$ 1 - 8iT - 59T^{2} $
	61	$ 1 + 9.19T + 61T^{2} $
	67	$ 1 - 13.1iT - 67T^{2} $
	71	$ 1 + 4.73iT - 71T^{2} $
	73	$ 1 - 6.26iT - 73T^{2} $
	79	$ 1 + 2.53T + 79T^{2} $
	83	$ 1 - 0.196iT - 83T^{2} $
	89	$ 1 + 9.46iT - 89T^{2} $
	97	$ 1 + 6iT - 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

−8.792680395151095991547839872991, −8.369666682555161815287256840895, −7.36610669259380630922397907730, −6.10515057302360738375249611550, −5.48116366257970987376170573365, −4.62555032332740810752014575760, −4.21112583507881965540923792629, −2.83745548447495961598720396639, −1.93811278977592849141006087175, −0.995480025414124806813537361840, 0.52473280173066694715008269147, 2.33278580484580706058714473882, 3.20462950018185532318714754843, 4.06429677307246762690050251045, 4.86197886534978035128148754285, 6.02871057900519190289925326956, 6.73317546619846246620502064922, 7.05639118133519292312589884447, 7.69297786696288945028542804143, 8.698372433675112799586084749494

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.387873465\)
\(L(\frac12)\)	\(\approx\)	\(1.387873465\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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