LMFDB - L-function 2-1008-21.20-c3-0-5

Dirichlet series

L(s) = 1

− 21.0·5^-s + (11 + 14.8i)7^-s + 15.5i·11^-s + 29.7i·13^-s + 63.2·17^-s + 89.3i·19^-s − 77.7i·23^-s + 318.·25^-s − 125. i·29^-s + 238. i·31^-s + (−231. − 313. i)35^-s − 184·37^-s + 105.·41^-s + 190·43^-s + 42.1·47^-s + ⋯

L(s) = 1

− 1.88·5^-s + (0.593 + 0.804i)7^-s + 0.426i·11^-s + 0.635i·13^-s + 0.901·17^-s + 1.07i·19^-s − 0.705i·23^-s + 2.55·25^-s − 0.805i·29^-s + 1.38i·31^-s + (−1.11 − 1.51i)35^-s − 0.817·37^-s + 0.401·41^-s + 0.673·43^-s + 0.130·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$-0.999 + 0.0204i$
Analytic conductor:	$59.4739$
Root analytic conductor:	$7.71193$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :3/2),\ -0.999 + 0.0204i)$

Particular Values

$L(2)$	$\approx$	$0.5756288942$
$L(\frac12)$	$\approx$	$0.5756288942$
$L(\frac{5}{2})$		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 $
	7	$ 1 + (-11 - 14.8i)T $
good	5	$ 1 + 21.0T + 125T^{2} $
	11	$ 1 - 15.5iT - 1.33e3T^{2} $
	13	$ 1 - 29.7iT - 2.19e3T^{2} $
	17	$ 1 - 63.2T + 4.91e3T^{2} $
	19	$ 1 - 89.3iT - 6.85e3T^{2} $
	23	$ 1 + 77.7iT - 1.21e4T^{2} $
	29	$ 1 + 125. iT - 2.43e4T^{2} $
	31	$ 1 - 238. iT - 2.97e4T^{2} $
	37	$ 1 + 184T + 5.06e4T^{2} $
	41	$ 1 - 105.T + 6.89e4T^{2} $
	43	$ 1 - 190T + 7.95e4T^{2} $
	47	$ 1 - 42.1T + 1.03e5T^{2} $
	53	$ 1 - 357. iT - 1.48e5T^{2} $
	59	$ 1 - 84.2T + 2.05e5T^{2} $
	61	$ 1 + 655. iT - 2.26e5T^{2} $
	67	$ 1 + 296T + 3.00e5T^{2} $
	71	$ 1 - 329. iT - 3.57e5T^{2} $
	73	$ 1 - 804. iT - 3.89e5T^{2} $
	79	$ 1 + 836T + 4.93e5T^{2} $
	83	$ 1 + 1.22e3T + 5.71e5T^{2} $
	89	$ 1 + 695.T + 7.04e5T^{2} $
	97	$ 1 + 566. iT - 9.12e5T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.06663803149199881769753788942, −8.888929884933184581883646309091, −8.274958533798591959796274567763, −7.64598220965650189916550685810, −6.84240309975801776018506888278, −5.60226913390946430457912466678, −4.58581311774775998305184631169, −3.90237207799357689496829756303, −2.82147345830056843226560733987, −1.36011212350590107593851550640, 0.18062066221630131497171129441, 1.06625760258022828552797566299, 3.01264545073106386118664557784, 3.79775957083581739833083276745, 4.57289845668700493157914744029, 5.54604821581197858399795919200, 7.03303589619805230688352029660, 7.53765529368854379669555148368, 8.142973053132720858472103604369, 8.934139178341758516863075396130

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.5756288942\)
\(L(\frac12)\)	\(\approx\)	\(0.5756288942\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more