LMFDB - L-function 2-1800-15.14-c2-0-23

Dirichlet series

L(s) = 1

− 3.16i·7^-s − 5.88i·11^-s + 1.16i·13^-s + 17.8·17^-s + 26.9·19^-s − 16.2·23^-s + 19.9i·29^-s − 29.6·31^-s + 15.4i·37^-s − 37.1i·41^-s + 8.27i·43^-s + 17.4·47^-s + 39·49^-s − 83.8·53^-s − 75.1i·59^-s + ⋯

L(s) = 1

− 0.451i·7^-s − 0.535i·11^-s + 0.0894i·13^-s + 1.05·17^-s + 1.41·19^-s − 0.706·23^-s + 0.689i·29^-s − 0.955·31^-s + 0.418i·37^-s − 0.907i·41^-s + 0.192i·43^-s + 0.370·47^-s + 0.795·49^-s − 1.58·53^-s − 1.27i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.472 + 0.881i$
Analytic conductor:	$49.0464$
Root analytic conductor:	$7.00331$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :1),\ 0.472 + 0.881i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.913430428$
$L(\frac12)$	$\approx$	$1.913430428$
$L(2)$		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 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 3.16iT - 49T^{2} $
	11	$ 1 + 5.88iT - 121T^{2} $
	13	$ 1 - 1.16iT - 169T^{2} $
	17	$ 1 - 17.8T + 289T^{2} $
	19	$ 1 - 26.9T + 361T^{2} $
	23	$ 1 + 16.2T + 529T^{2} $
	29	$ 1 - 19.9iT - 841T^{2} $
	31	$ 1 + 29.6T + 961T^{2} $
	37	$ 1 - 15.4iT - 1.36e3T^{2} $
	41	$ 1 + 37.1iT - 1.68e3T^{2} $
	43	$ 1 - 8.27iT - 1.84e3T^{2} $
	47	$ 1 - 17.4T + 2.20e3T^{2} $
	53	$ 1 + 83.8T + 2.80e3T^{2} $
	59	$ 1 + 75.1iT - 3.48e3T^{2} $
	61	$ 1 - 82.2T + 3.72e3T^{2} $
	67	$ 1 + 79.8iT - 4.48e3T^{2} $
	71	$ 1 + 24.0iT - 5.04e3T^{2} $
	73	$ 1 + 107. iT - 5.32e3T^{2} $
	79	$ 1 - 62.3T + 6.24e3T^{2} $
	83	$ 1 - 68.7T + 6.88e3T^{2} $
	89	$ 1 + 86.7iT - 7.92e3T^{2} $
	97	$ 1 - 99.5iT - 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

−9.063537688825542346423198663203, −7.992555130045663244844046345774, −7.52323984656031250048554984935, −6.59125493854834458907922306979, −5.65021984071307193690941926580, −4.98998292867937279373710781880, −3.74219095921925770865429838371, −3.16970254625750426849002629333, −1.72811836467146292794236484024, −0.59009614583986417322468118091, 1.04978784890635986858540014901, 2.26036480999434453775639502784, 3.28310693262232162294190456043, 4.22801002866759863651309954818, 5.36257169666305957801428319973, 5.80321737967633688212690008625, 6.97381258876268597554526873469, 7.66177286510883695858862079269, 8.355126565515831727740674183964, 9.444468503109753371506038261284

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

\(L(\frac{3}{2})\)	\(\approx\)	\(1.913430428\)
\(L(\frac12)\)	\(\approx\)	\(1.913430428\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more