LMFDB - L-function 2-1216-152.75-c1-0-30

Dirichlet series

L(s) = 1

− 1.12i·3^-s − 0.0952i·7^-s + 1.72·9^-s − 6.35·13^-s − 4.53·17^-s − 4.35i·19^-s − 0.107·21^-s − 9.35i·23^-s + 5·25^-s − 5.33i·27^-s + 4.09·29^-s − 8.71·37^-s + 7.16i·39^-s + 6i·47^-s + 6.99·49^-s + ⋯

L(s) = 1

− 0.651i·3^-s − 0.0360i·7^-s + 0.575·9^-s − 1.76·13^-s − 1.10·17^-s − 0.999i·19^-s − 0.0234·21^-s − 1.95i·23^-s + 25^-s − 1.02i·27^-s + 0.760·29^-s − 1.43·37^-s + 1.14i·39^-s + 0.875i·47^-s + 0.998·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$9.70980$
Root analytic conductor:	$3.11605$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.064694116$
$L(\frac12)$	$\approx$	$1.064694116$
$L(\frac{3}{2})$		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 + 4.35iT $
good	3	$ 1 + 1.12iT - 3T^{2} $
	5	$ 1 - 5T^{2} $
	7	$ 1 + 0.0952iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 6.35T + 13T^{2} $
	17	$ 1 + 4.53T + 17T^{2} $
	23	$ 1 + 9.35iT - 23T^{2} $
	29	$ 1 - 4.09T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 8.71T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 - 6iT - 47T^{2} $
	53	$ 1 + 10.8T + 53T^{2} $
	59	$ 1 + 11.5iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 16.0iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 13.8T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 97T^{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

−9.362735019357545962361562239015, −8.607355216930506665133940322702, −7.64474025816204303275052323347, −6.84317014016890899346763816085, −6.49628602024531548875605615952, −4.83829542002361841876870225752, −4.57259435426885155160202696732, −2.85352720144259483705445487312, −2.05308892410715951515805431451, −0.43552011424824182956852510360, 1.69041739654494811522173149503, 2.97389607941044434835533942242, 4.08289636517113035491725451611, 4.85544481538038622343019883794, 5.62649594063522801944203624348, 6.98644125097223400792259370448, 7.37471849955982027788965262250, 8.575011811298677309387275420656, 9.359664457563982274848133797782, 10.06593346258000413553414653428

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.064694116\)
\(L(\frac12)\)	\(\approx\)	\(1.064694116\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more