LMFDB - L-function 2-38e2-19.18-c2-0-28

Dirichlet series

L(s) = 1

+ 4.89i·3^-s − 9.07·5^-s + 11.2·7^-s − 14.9·9^-s − 5.07·11^-s + 2.03i·13^-s − 44.3i·15^-s − 15.6·17^-s + 54.9i·21^-s − 19.4·23^-s + 57.3·25^-s − 29.0i·27^-s − 1.45i·29^-s − 44.7i·31^-s − 24.8i·33^-s + ⋯

L(s) = 1

+ 1.63i·3^-s − 1.81·5^-s + 1.60·7^-s − 1.65·9^-s − 0.461·11^-s + 0.156i·13^-s − 2.95i·15^-s − 0.918·17^-s + 2.61i·21^-s − 0.846·23^-s + 2.29·25^-s − 1.07i·27^-s − 0.0502i·29^-s − 1.44i·31^-s − 0.752i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1444$ = $2^{2} \cdot 19^{2}$
Sign:	$0.989 + 0.143i$
Analytic conductor:	$39.3461$
Root analytic conductor:	$6.27265$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1444} (721, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1444,\ (\ :1),\ 0.989 + 0.143i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.8086745508$
$L(\frac12)$	$\approx$	$0.8086745508$
$L(2)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	19	$ 1 $
good	3	$ 1 - 4.89iT - 9T^{2} $
	5	$ 1 + 9.07T + 25T^{2} $
	7	$ 1 - 11.2T + 49T^{2} $
	11	$ 1 + 5.07T + 121T^{2} $
	13	$ 1 - 2.03iT - 169T^{2} $
	17	$ 1 + 15.6T + 289T^{2} $
	23	$ 1 + 19.4T + 529T^{2} $
	29	$ 1 + 1.45iT - 841T^{2} $
	31	$ 1 + 44.7iT - 961T^{2} $
	37	$ 1 + 34.8iT - 1.36e3T^{2} $
	41	$ 1 + 60.8iT - 1.68e3T^{2} $
	43	$ 1 - 11.1T + 1.84e3T^{2} $
	47	$ 1 + 14.7T + 2.20e3T^{2} $
	53	$ 1 + 16.7iT - 2.80e3T^{2} $
	59	$ 1 + 6.20iT - 3.48e3T^{2} $
	61	$ 1 - 71.3T + 3.72e3T^{2} $
	67	$ 1 - 65.0iT - 4.48e3T^{2} $
	71	$ 1 - 108. iT - 5.04e3T^{2} $
	73	$ 1 + 35.3T + 5.32e3T^{2} $
	79	$ 1 + 113. iT - 6.24e3T^{2} $
	83	$ 1 - 28.8T + 6.88e3T^{2} $
	89	$ 1 - 60.0iT - 7.92e3T^{2} $
	97	$ 1 + 29.7iT - 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

−9.185291017261201620727107761490, −8.489188296200656114873145031186, −7.956194543008641596548523414852, −7.19721568009431285044686997203, −5.60888437820566968757394386119, −4.79079371306550958438383145790, −4.18113931984474532868184631361, −3.77068252997219807261367452086, −2.34398874428887168617740268716, −0.29045826522517544284091588263, 0.912958840694429652607646910176, 1.93060187060725162724717272923, 3.10667732470353044524823474221, 4.38678697871003321769613209073, 5.06301005297739686182742894691, 6.39940390930868852077387812022, 7.19131272725382987898194406598, 7.84963645310083623672849355931, 8.181329431522899053584716002420, 8.746664226076100318424494652573

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.8086745508\)
\(L(\frac12)\)	\(\approx\)	\(0.8086745508\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more