LMFDB - L-function 2-1344-8.5-c3-0-45

Dirichlet series

L(s) = 1

− 3i·3^-s − 8.40i·5^-s + 7·7^-s − 9·9^-s − 42.3i·11^-s + 9.41i·13^-s − 25.2·15^-s + 107.·17^-s + 124. i·19^-s − 21i·21^-s + 55.2·23^-s + 54.2·25^-s + 27i·27^-s + 47.1i·29^-s + 318.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.752i·5^-s + 0.377·7^-s − 0.333·9^-s − 1.15i·11^-s + 0.200i·13^-s − 0.434·15^-s + 1.53·17^-s + 1.50i·19^-s − 0.218i·21^-s + 0.501·23^-s + 0.434·25^-s + 0.192i·27^-s + 0.301i·29^-s + 1.84·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.258 + 0.965i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (673, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.258 + 0.965i)$

Particular Values

$L(2)$	$\approx$	$2.524233925$
$L(\frac12)$	$\approx$	$2.524233925$
$L(\frac{5}{2})$		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 + 3iT $
	7	$ 1 - 7T $
good	5	$ 1 + 8.40iT - 125T^{2} $
	11	$ 1 + 42.3iT - 1.33e3T^{2} $
	13	$ 1 - 9.41iT - 2.19e3T^{2} $
	17	$ 1 - 107.T + 4.91e3T^{2} $
	19	$ 1 - 124. iT - 6.85e3T^{2} $
	23	$ 1 - 55.2T + 1.21e4T^{2} $
	29	$ 1 - 47.1iT - 2.43e4T^{2} $
	31	$ 1 - 318.T + 2.97e4T^{2} $
	37	$ 1 + 88.8iT - 5.06e4T^{2} $
	41	$ 1 - 105.T + 6.89e4T^{2} $
	43	$ 1 - 202. iT - 7.95e4T^{2} $
	47	$ 1 - 176.T + 1.03e5T^{2} $
	53	$ 1 + 393. iT - 1.48e5T^{2} $
	59	$ 1 + 898. iT - 2.05e5T^{2} $
	61	$ 1 - 904. iT - 2.26e5T^{2} $
	67	$ 1 + 67.0iT - 3.00e5T^{2} $
	71	$ 1 + 620.T + 3.57e5T^{2} $
	73	$ 1 - 481.T + 3.89e5T^{2} $
	79	$ 1 + 375.T + 4.93e5T^{2} $
	83	$ 1 - 976. iT - 5.71e5T^{2} $
	89	$ 1 - 1.18e3T + 7.04e5T^{2} $
	97	$ 1 - 599.T + 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

−8.856823803430912573448896268640, −8.186223028038888637091471700102, −7.72389442186080051546502166570, −6.50875424897428529416447034397, −5.72646000747331818121794575798, −5.03222718615740285848928828398, −3.82244497363512717695046788560, −2.85391134842066402769687717767, −1.40511751500003690596048810093, −0.792994280111450543561968971914, 0.947042220458243363455875302021, 2.47235818913426345247214997296, 3.18782458865649685283284798397, 4.43595627080263316732517088075, 5.02625657990850013197353282144, 6.10593396094574413235157258438, 7.07729324081811599349508828578, 7.66086037559597179420391339959, 8.694501074682207568809652400625, 9.541988126289224515981458749222

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.524233925\)
\(L(\frac12)\)	\(\approx\)	\(2.524233925\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more