LMFDB - L-function 2-48e2-24.11-c1-0-28

Dirichlet series

L(s) = 1

− 1.41·5^-s − 4i·7^-s − 5.65i·11^-s + 4i·13^-s − 4.24i·17^-s + 5.65·23^-s − 2.99·25^-s − 1.41·29^-s + 4i·31^-s + 5.65i·35^-s + 6i·37^-s − 9.89i·41^-s − 8·43^-s − 5.65·47^-s − 9·49^-s + ⋯

L(s) = 1

− 0.632·5^-s − 1.51i·7^-s − 1.70i·11^-s + 1.10i·13^-s − 1.02i·17^-s + 1.17·23^-s − 0.599·25^-s − 0.262·29^-s + 0.718i·31^-s + 0.956i·35^-s + 0.986i·37^-s − 1.54i·41^-s − 1.21·43^-s − 0.825·47^-s − 1.28·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$-0.985 + 0.169i$
Analytic conductor:	$18.3975$
Root analytic conductor:	$4.28923$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1/2),\ -0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$0.7566799371$
$L(\frac12)$	$\approx$	$0.7566799371$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
good	5	$ 1 + 1.41T + 5T^{2} $
	7	$ 1 + 4iT - 7T^{2} $
	11	$ 1 + 5.65iT - 11T^{2} $
	13	$ 1 - 4iT - 13T^{2} $
	17	$ 1 + 4.24iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 - 5.65T + 23T^{2} $
	29	$ 1 + 1.41T + 29T^{2} $
	31	$ 1 - 4iT - 31T^{2} $
	37	$ 1 - 6iT - 37T^{2} $
	41	$ 1 + 9.89iT - 41T^{2} $
	43	$ 1 + 8T + 43T^{2} $
	47	$ 1 + 5.65T + 47T^{2} $
	53	$ 1 + 4.24T + 53T^{2} $
	59	$ 1 - 11.3iT - 59T^{2} $
	61	$ 1 + 2iT - 61T^{2} $
	67	$ 1 - 8T + 67T^{2} $
	71	$ 1 + 5.65T + 71T^{2} $
	73	$ 1 + 73T^{2} $
	79	$ 1 + 4iT - 79T^{2} $
	83	$ 1 + 5.65iT - 83T^{2} $
	89	$ 1 + 4.24iT - 89T^{2} $
	97	$ 1 + 8T + 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

−8.612587953888344605911617311777, −7.83124197310351650918350577215, −7.04456831668041609528843411747, −6.59166697393143525298857321394, −5.38125373658388259581740386414, −4.49501941536508183947261119690, −3.69566567629417854955365174138, −3.04380700840026546236922125318, −1.33405488899164955820269264399, −0.26865019386898115791130679901, 1.71218191141682143190520481053, 2.63321674541532186960573710080, 3.62832762016046146188646819198, 4.69417469519543643888693367800, 5.36985728857996890638452918503, 6.21425239347208837133084304307, 7.12176969982420803632771652233, 8.013166026067236912029835273701, 8.397009462903708334173222812517, 9.515955599084478855586441896775

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7566799371\)
\(L(\frac12)\)	\(\approx\)	\(0.7566799371\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more