LMFDB - L-function 2-33e2-3.2-c2-0-46

Dirichlet series

L(s) = 1

− 3.89i·2^-s − 11.2·4^-s + 6.10i·5^-s − 2.61·7^-s + 28.0i·8^-s + 23.7·10^-s + 7.69·13^-s + 10.1i·14^-s + 64.7·16^-s − 27.5i·17^-s − 3.63·19^-s − 68.3i·20^-s + 22.4i·23^-s − 12.2·25^-s − 30.0i·26^-s + ⋯

L(s) = 1

− 1.94i·2^-s − 2.80·4^-s + 1.22i·5^-s − 0.372·7^-s + 3.51i·8^-s + 2.37·10^-s + 0.592·13^-s + 0.727i·14^-s + 4.04·16^-s − 1.62i·17^-s − 0.191·19^-s − 3.41i·20^-s + 0.975i·23^-s − 0.490·25^-s − 1.15i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$-0.577 - 0.816i$
Analytic conductor:	$29.6731$
Root analytic conductor:	$5.44730$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1089} (485, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1089,\ (\ :1),\ -0.577 - 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5184967051$
$L(\frac12)$	$\approx$	$0.5184967051$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	11	$ 1 $
good	2	$ 1 + 3.89iT - 4T^{2} $
	5	$ 1 - 6.10iT - 25T^{2} $
	7	$ 1 + 2.61T + 49T^{2} $
	13	$ 1 - 7.69T + 169T^{2} $
	17	$ 1 + 27.5iT - 289T^{2} $
	19	$ 1 + 3.63T + 361T^{2} $
	23	$ 1 - 22.4iT - 529T^{2} $
	29	$ 1 + 16.9iT - 841T^{2} $
	31	$ 1 - 3.26T + 961T^{2} $
	37	$ 1 - 15.0T + 1.36e3T^{2} $
	41	$ 1 - 40.2iT - 1.68e3T^{2} $
	43	$ 1 + 69.4T + 1.84e3T^{2} $
	47	$ 1 + 21.7iT - 2.20e3T^{2} $
	53	$ 1 + 12.1iT - 2.80e3T^{2} $
	59	$ 1 + 34.0iT - 3.48e3T^{2} $
	61	$ 1 + 61.3T + 3.72e3T^{2} $
	67	$ 1 - 54.6T + 4.48e3T^{2} $
	71	$ 1 - 11.5iT - 5.04e3T^{2} $
	73	$ 1 + 41.7T + 5.32e3T^{2} $
	79	$ 1 + 96.9T + 6.24e3T^{2} $
	83	$ 1 + 89.8iT - 6.88e3T^{2} $
	89	$ 1 + 117. iT - 7.92e3T^{2} $
	97	$ 1 + 7.47T + 9.40e3T^{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

−9.688998286250362854666744891759, −8.696896960046656201685536549613, −7.69006467232917383803400727252, −6.55912251825920171322677240251, −5.37273713130931360535256127938, −4.33746295835283035632688425577, −3.22232077395230589328649847367, −2.89714364048054239135845832980, −1.65702288041194896771742927732, −0.18450046775317236672999191910, 1.18186370540942317942088562555, 3.68588437896094974569884694644, 4.47416918086553670004175002233, 5.27659960456947664421171138350, 6.12309714463546461768846161445, 6.68537825253396111180700329120, 7.83593477921887147357248024134, 8.528732597838042767823621059896, 8.839244475554597449414301923188, 9.778468891201717909010170053778

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5184967051\)
\(L(\frac12)\)	\(\approx\)	\(0.5184967051\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more