LMFDB - L-function 2-12e3-24.11-c3-0-13

Dirichlet series

L(s) = 1

− 16.5·5^-s − 0.800i·7^-s + 51.7i·11^-s − 83.0i·13^-s − 20.1i·17^-s + 22.3·19^-s − 79.4·23^-s + 147.·25^-s − 134.·29^-s + 33.5i·31^-s + 13.2i·35^-s − 384. i·37^-s − 71.4i·41^-s − 464.·43^-s + 278.·47^-s + ⋯

L(s) = 1

− 1.47·5^-s − 0.0432i·7^-s + 1.41i·11^-s − 1.77i·13^-s − 0.287i·17^-s + 0.269·19^-s − 0.719·23^-s + 1.17·25^-s − 0.861·29^-s + 0.194i·31^-s + 0.0637i·35^-s − 1.70i·37^-s − 0.272i·41^-s − 1.64·43^-s + 0.863·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.258 - 0.965i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 0.258 - 0.965i)$

Particular Values

$L(2)$	$\approx$	$0.6870257960$
$L(\frac12)$	$\approx$	$0.6870257960$
$L(\frac{5}{2})$		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 $
good	5	$ 1 + 16.5T + 125T^{2} $
	7	$ 1 + 0.800iT - 343T^{2} $
	11	$ 1 - 51.7iT - 1.33e3T^{2} $
	13	$ 1 + 83.0iT - 2.19e3T^{2} $
	17	$ 1 + 20.1iT - 4.91e3T^{2} $
	19	$ 1 - 22.3T + 6.85e3T^{2} $
	23	$ 1 + 79.4T + 1.21e4T^{2} $
	29	$ 1 + 134.T + 2.43e4T^{2} $
	31	$ 1 - 33.5iT - 2.97e4T^{2} $
	37	$ 1 + 384. iT - 5.06e4T^{2} $
	41	$ 1 + 71.4iT - 6.89e4T^{2} $
	43	$ 1 + 464.T + 7.95e4T^{2} $
	47	$ 1 - 278.T + 1.03e5T^{2} $
	53	$ 1 - 362.T + 1.48e5T^{2} $
	59	$ 1 + 315. iT - 2.05e5T^{2} $
	61	$ 1 + 126. iT - 2.26e5T^{2} $
	67	$ 1 + 857.T + 3.00e5T^{2} $
	71	$ 1 - 771.T + 3.57e5T^{2} $
	73	$ 1 + 687.T + 3.89e5T^{2} $
	79	$ 1 - 317. iT - 4.93e5T^{2} $
	83	$ 1 - 184. iT - 5.71e5T^{2} $
	89	$ 1 + 1.26e3iT - 7.04e5T^{2} $
	97	$ 1 - 871.T + 9.12e5T^{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

−9.028833595592119684611646171741, −8.137800244076937221603660740165, −7.48859795940332394218907038517, −7.15564169002863598276138218384, −5.77481475004176761722888370091, −4.95621059009804543861596075126, −4.03987957533559349041642194798, −3.36222113430489067312517372736, −2.18687136074067850294999929242, −0.67152076617546061102243229717, 0.23287200828208987198457424597, 1.52200913812968683533916549871, 2.96202576308189609788437830186, 3.86667803871648362787482199720, 4.36380811543127401723643253884, 5.56859579773490673465154174417, 6.51372871770729664701808319189, 7.23762023010912568837641059688, 8.134547310312557771883846190742, 8.615854253123468342187289789385

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(0.6870257960\)
\(L(\frac12)\)	\(\approx\)	\(0.6870257960\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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