LMFDB - L-function 2-608-19.18-c2-0-31

Dirichlet series

L(s) = 1

− 0.682i·3^-s + 6.57·5^-s + 8.44·7^-s + 8.53·9^-s + 2.26·11^-s − 19.5i·13^-s − 4.48i·15^-s + 7.76·17^-s + (−7.24 + 17.5i)19^-s − 5.76i·21^-s − 27.5·23^-s + 18.2·25^-s − 11.9i·27^-s + 48.3i·29^-s + 18.1i·31^-s + ⋯

L(s) = 1

− 0.227i·3^-s + 1.31·5^-s + 1.20·7^-s + 0.948·9^-s + 0.205·11^-s − 1.50i·13^-s − 0.299i·15^-s + 0.456·17^-s + (−0.381 + 0.924i)19^-s − 0.274i·21^-s − 1.19·23^-s + 0.729·25^-s − 0.443i·27^-s + 1.66i·29^-s + 0.586i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$608$ = $2^{5} \cdot 19$
Sign:	$0.924 + 0.381i$
Analytic conductor:	$16.5668$
Root analytic conductor:	$4.07023$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{608} (417, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 608,\ (\ :1),\ 0.924 + 0.381i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.808933527$
$L(\frac12)$	$\approx$	$2.808933527$
$L(2)$		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 + (7.24 - 17.5i)T $
good	3	$ 1 + 0.682iT - 9T^{2} $
	5	$ 1 - 6.57T + 25T^{2} $
	7	$ 1 - 8.44T + 49T^{2} $
	11	$ 1 - 2.26T + 121T^{2} $
	13	$ 1 + 19.5iT - 169T^{2} $
	17	$ 1 - 7.76T + 289T^{2} $
	23	$ 1 + 27.5T + 529T^{2} $
	29	$ 1 - 48.3iT - 841T^{2} $
	31	$ 1 - 18.1iT - 961T^{2} $
	37	$ 1 + 42.8iT - 1.36e3T^{2} $
	41	$ 1 + 19.0iT - 1.68e3T^{2} $
	43	$ 1 + 74.5T + 1.84e3T^{2} $
	47	$ 1 - 38.1T + 2.20e3T^{2} $
	53	$ 1 + 8.32iT - 2.80e3T^{2} $
	59	$ 1 + 66.1iT - 3.48e3T^{2} $
	61	$ 1 - 50.5T + 3.72e3T^{2} $
	67	$ 1 + 69.4iT - 4.48e3T^{2} $
	71	$ 1 - 48.1iT - 5.04e3T^{2} $
	73	$ 1 - 71.7T + 5.32e3T^{2} $
	79	$ 1 - 147. iT - 6.24e3T^{2} $
	83	$ 1 + 103.T + 6.88e3T^{2} $
	89	$ 1 - 143. iT - 7.92e3T^{2} $
	97	$ 1 + 49.3iT - 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

−10.27548701838307969051895852591, −9.776578400129348684968676716997, −8.498037305896028714018119207384, −7.83406435385971833292934848943, −6.78040384789597486891115402493, −5.66403563237651695896969562574, −5.10183981294612074684296469816, −3.69766566378379783615603651928, −2.08134643207869606898978162288, −1.30676262098896211302628973242, 1.51026513745120956345211331146, 2.21555816351295834064360031185, 4.16799342300059875991516467312, 4.80481272600911327312374357070, 5.98170548128520259593417877362, 6.79545384357501587571442735317, 7.889662691951327838883691591145, 8.909586262755543636630522111054, 9.767034748277112204978316574424, 10.22760196885573544947824356191

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(2.808933527\)
\(L(\frac12)\)	\(\approx\)	\(2.808933527\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more