LMFDB - L-function 2-15e2-5.4-c9-0-29

Dirichlet series

L(s) = 1

+ 4i·2^-s + 496·4^-s − 7.68e3i·7^-s + 4.03e3i·8^-s + 8.64e4·11^-s + 1.49e5i·13^-s + 3.07e4·14^-s + 2.37e5·16^-s + 2.07e5i·17^-s − 7.16e5·19^-s + 3.45e5i·22^-s + 1.36e6i·23^-s − 5.99e5·26^-s − 3.80e6i·28^-s − 3.19e6·29^-s + ⋯

L(s) = 1

+ 0.176i·2^-s + 0.968·4^-s − 1.20i·7^-s + 0.348i·8^-s + 1.77·11^-s + 1.45i·13^-s + 0.213·14^-s + 0.907·16^-s + 0.602i·17^-s − 1.26·19^-s + 0.314i·22^-s + 1.02i·23^-s − 0.257·26^-s − 1.17i·28^-s − 0.838·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$115.883$
Root analytic conductor:	$10.7648$
Motivic weight:	$9$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :9/2),\ 0.447 - 0.894i)$

Particular Values

$L(5)$	$\approx$	$3.044681655$
$L(\frac12)$	$\approx$	$3.044681655$
$L(\frac{11}{2})$		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 $
	5	$ 1 $
good	2	$ 1 - 4iT - 512T^{2} $
	7	$ 1 + 7.68e3iT - 4.03e7T^{2} $
	11	$ 1 - 8.64e4T + 2.35e9T^{2} $
	13	$ 1 - 1.49e5iT - 1.06e10T^{2} $
	17	$ 1 - 2.07e5iT - 1.18e11T^{2} $
	19	$ 1 + 7.16e5T + 3.22e11T^{2} $
	23	$ 1 - 1.36e6iT - 1.80e12T^{2} $
	29	$ 1 + 3.19e6T + 1.45e13T^{2} $
	31	$ 1 + 2.34e6T + 2.64e13T^{2} $
	37	$ 1 - 1.87e7iT - 1.29e14T^{2} $
	41	$ 1 - 2.92e7T + 3.27e14T^{2} $
	43	$ 1 - 1.51e6iT - 5.02e14T^{2} $
	47	$ 1 + 6.15e5iT - 1.11e15T^{2} $
	53	$ 1 - 4.74e6iT - 3.29e15T^{2} $
	59	$ 1 - 6.06e7T + 8.66e15T^{2} $
	61	$ 1 + 1.26e8T + 1.16e16T^{2} $
	67	$ 1 + 1.11e8iT - 2.72e16T^{2} $
	71	$ 1 - 1.75e8T + 4.58e16T^{2} $
	73	$ 1 - 6.12e7iT - 5.88e16T^{2} $
	79	$ 1 + 2.34e8T + 1.19e17T^{2} $
	83	$ 1 - 1.18e8iT - 1.86e17T^{2} $
	89	$ 1 + 3.16e8T + 3.50e17T^{2} $
	97	$ 1 - 2.42e8iT - 7.60e17T^{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

−11.02446512781336997865724112360, −9.853406882625657095197250942862, −8.852772693681937122748731535332, −7.54304514544516979353807668168, −6.74352036254651512650312980855, −6.15445857889417270276188540624, −4.32044408965345829776686262395, −3.64456909902492691892661165326, −1.89963845105653498786930770886, −1.21114987802860942644848370986, 0.61148901169788849291609049794, 1.92838684942071060986755102368, 2.78249443332627109442736792174, 3.99721471299958580141394183352, 5.63248541578336711116012634397, 6.29691999929754003733322518494, 7.37942136923554637505514196724, 8.600707122103742163632789795270, 9.413418267996648451049760721358, 10.65981164474762885754600570163

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(3.044681655\)
\(L(\frac12)\)	\(\approx\)	\(3.044681655\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more