LMFDB - L-function 2-2240-28.27-c1-0-42

Dirichlet series

L(s) = 1

− 1.77·3^-s − i·5^-s + (0.829 − 2.51i)7^-s + 0.148·9^-s + 6.36i·11^-s − 0.563i·13^-s + 1.77i·15^-s − 6.12i·17^-s + 0.694·19^-s + (−1.47 + 4.45i)21^-s + 0.964i·23^-s − 25^-s + 5.05·27^-s + 2.02·29^-s + 8.09·31^-s + ⋯

L(s) = 1

− 1.02·3^-s − 0.447i·5^-s + (0.313 − 0.949i)7^-s + 0.0494·9^-s + 1.91i·11^-s − 0.156i·13^-s + 0.458i·15^-s − 1.48i·17^-s + 0.159·19^-s + (−0.321 + 0.972i)21^-s + 0.201i·23^-s − 0.200·25^-s + 0.973·27^-s + 0.376·29^-s + 1.45·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.313 + 0.949i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (1791, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ -0.313 + 0.949i)$

Particular Values

$L(1)$	$\approx$	$0.8386261605$
$L(\frac12)$	$\approx$	$0.8386261605$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
	7	$ 1 + (-0.829 + 2.51i)T $
good	3	$ 1 + 1.77T + 3T^{2} $
	11	$ 1 - 6.36iT - 11T^{2} $
	13	$ 1 + 0.563iT - 13T^{2} $
	17	$ 1 + 6.12iT - 17T^{2} $
	19	$ 1 - 0.694T + 19T^{2} $
	23	$ 1 - 0.964iT - 23T^{2} $
	29	$ 1 - 2.02T + 29T^{2} $
	31	$ 1 - 8.09T + 31T^{2} $
	37	$ 1 + 1.89T + 37T^{2} $
	41	$ 1 - 1.97iT - 41T^{2} $
	43	$ 1 + 5.26iT - 43T^{2} $
	47	$ 1 - 3.03T + 47T^{2} $
	53	$ 1 + 1.73T + 53T^{2} $
	59	$ 1 + 12.9T + 59T^{2} $
	61	$ 1 + 9.96iT - 61T^{2} $
	67	$ 1 - 0.714iT - 67T^{2} $
	71	$ 1 - 7.90iT - 71T^{2} $
	73	$ 1 - 0.743iT - 73T^{2} $
	79	$ 1 - 3.94iT - 79T^{2} $
	83	$ 1 + 11.5T + 83T^{2} $
	89	$ 1 + 16.6iT - 89T^{2} $
	97	$ 1 + 9.54iT - 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

−8.859843055980412430615588224277, −7.81748489751975940830317909314, −7.17080764160690059490591900161, −6.58670198550954909879282893695, −5.46175402292902106373625226981, −4.72812706851745185374730247278, −4.39227433675859297045558109084, −2.88910898037024863004380211620, −1.54875234085893618179358781738, −0.39031452643896260451263020652, 1.11809627747971997862786207018, 2.59472428328683786903690637195, 3.43408281295809610701671660872, 4.63788390496825666799164858450, 5.58643088024683763075599179472, 6.12945398318802490162407856265, 6.46455917090947024588069310398, 7.915831310488168923365231975534, 8.467593621356619989122832878902, 9.099719684746167918680242524251

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8386261605\)
\(L(\frac12)\)	\(\approx\)	\(0.8386261605\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more