LMFDB - L-function 2-3087-21.20-c1-0-69

Dirichlet series

L(s) = 1

− 0.660i·2^-s + 1.56·4^-s + 3.37·5^-s − 2.35i·8^-s − 2.23i·10^-s + 0.0311i·11^-s + 1.36i·13^-s + 1.57·16^-s + 0.499·17^-s − 8.07i·19^-s + 5.28·20^-s + 0.0205·22^-s + 4.60i·23^-s + 6.40·25^-s + 0.900·26^-s + ⋯

L(s) = 1

− 0.467i·2^-s + 0.781·4^-s + 1.51·5^-s − 0.832i·8^-s − 0.705i·10^-s + 0.00939i·11^-s + 0.377i·13^-s + 0.393·16^-s + 0.121·17^-s − 1.85i·19^-s + 1.18·20^-s + 0.00439·22^-s + 0.960i·23^-s + 1.28·25^-s + 0.176·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3087$ = $3^{2} \cdot 7^{3}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$24.6498$
Root analytic conductor:	$4.96485$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3087} (3086, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3087,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$3.256316360$
$L(\frac12)$	$\approx$	$3.256316360$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 $
good	2	$ 1 + 0.660iT - 2T^{2} $
	5	$ 1 - 3.37T + 5T^{2} $
	11	$ 1 - 0.0311iT - 11T^{2} $
	13	$ 1 - 1.36iT - 13T^{2} $
	17	$ 1 - 0.499T + 17T^{2} $
	19	$ 1 + 8.07iT - 19T^{2} $
	23	$ 1 - 4.60iT - 23T^{2} $
	29	$ 1 + 1.74iT - 29T^{2} $
	31	$ 1 - 5.76iT - 31T^{2} $
	37	$ 1 - 6.29T + 37T^{2} $
	41	$ 1 + 10.5T + 41T^{2} $
	43	$ 1 - 6.83T + 43T^{2} $
	47	$ 1 - 1.86T + 47T^{2} $
	53	$ 1 + 11.7iT - 53T^{2} $
	59	$ 1 - 8.79T + 59T^{2} $
	61	$ 1 - 4.53iT - 61T^{2} $
	67	$ 1 - 2.05T + 67T^{2} $
	71	$ 1 + 6.15iT - 71T^{2} $
	73	$ 1 + 13.0iT - 73T^{2} $
	79	$ 1 + 2.05T + 79T^{2} $
	83	$ 1 + 3.34T + 83T^{2} $
	89	$ 1 - 0.198T + 89T^{2} $
	97	$ 1 - 14.0iT - 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

−8.920991424611032371448859991386, −7.72982449958504395410309723754, −6.82662986062160428472829873209, −6.47417740845288268341165951760, −5.55149569076420383077545601127, −4.87054862976181800016896975486, −3.58698499753219697916108834981, −2.63587235808796394124984964111, −2.03464189755095918452229197959, −1.06344758921943941510658696782, 1.34397244440738159066119786075, 2.19099855415845039857809187488, 2.94442511066549771969351590962, 4.20438192705609445052903095495, 5.48685199171504428745124161136, 5.77873682885755656072281331040, 6.41384169972522705704898998860, 7.20233557947979583736338946967, 8.055542273901908583147565304136, 8.651656625277440477525607608185

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(3.256316360\)
\(L(\frac12)\)	\(\approx\)	\(3.256316360\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more