LMFDB - L-function 2-1452-11.10-c2-0-28

Dirichlet series

L(s) = 1

− 1.73·3^-s + 3.73·5^-s − 2.44i·7^-s + 2.99·9^-s − 14.5i·13^-s − 6.46·15^-s − 8.52i·17^-s − 2.07i·19^-s + 4.24i·21^-s + 21.5·23^-s − 11.0·25^-s − 5.19·27^-s + 18.4i·29^-s − 15.4·31^-s − 9.14i·35^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.746·5^-s − 0.349i·7^-s + 0.333·9^-s − 1.11i·13^-s − 0.430·15^-s − 0.501i·17^-s − 0.108i·19^-s + 0.202i·21^-s + 0.935·23^-s − 0.442·25^-s − 0.192·27^-s + 0.635i·29^-s − 0.498·31^-s − 0.261i·35^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1452$ = $2^{2} \cdot 3 \cdot 11^{2}$
Sign:	$-0.522 + 0.852i$
Analytic conductor:	$39.5641$
Root analytic conductor:	$6.29000$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1452} (241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1452,\ (\ :1),\ -0.522 + 0.852i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.176436764$
$L(\frac12)$	$\approx$	$1.176436764$
$L(2)$		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 + 1.73T $
	11	$ 1 $
good	5	$ 1 - 3.73T + 25T^{2} $
	7	$ 1 + 2.44iT - 49T^{2} $
	13	$ 1 + 14.5iT - 169T^{2} $
	17	$ 1 + 8.52iT - 289T^{2} $
	19	$ 1 + 2.07iT - 361T^{2} $
	23	$ 1 - 21.5T + 529T^{2} $
	29	$ 1 - 18.4iT - 841T^{2} $
	31	$ 1 + 15.4T + 961T^{2} $
	37	$ 1 + 52.5T + 1.36e3T^{2} $
	41	$ 1 + 67.3iT - 1.68e3T^{2} $
	43	$ 1 - 4.24iT - 1.84e3T^{2} $
	47	$ 1 + 31.4T + 2.20e3T^{2} $
	53	$ 1 + 17T + 2.80e3T^{2} $
	59	$ 1 + 22.1T + 3.48e3T^{2} $
	61	$ 1 - 17.7iT - 3.72e3T^{2} $
	67	$ 1 - 63.0T + 4.48e3T^{2} $
	71	$ 1 - 82.7T + 5.04e3T^{2} $
	73	$ 1 + 86.4iT - 5.32e3T^{2} $
	79	$ 1 - 37.1iT - 6.24e3T^{2} $
	83	$ 1 - 146. iT - 6.88e3T^{2} $
	89	$ 1 + 87.8T + 7.92e3T^{2} $
	97	$ 1 + 116.T + 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

−9.213102768166812890613398326490, −8.265292375831388586953654129490, −7.25009583286441719765613798713, −6.67374541773856960476642476795, −5.45530403663774233659841688801, −5.28630174543546313287049418666, −3.93283322369480774139257119939, −2.85936843219657742042262553243, −1.58428569915733268943107981425, −0.35417987608607435019794287294, 1.38611471180471224140654389124, 2.30162101008569273549081938323, 3.66861713521579888290464364385, 4.72689896711353028511861456087, 5.52636803478508061989757994384, 6.32615571290899030571525602440, 6.92654708643406183384920251777, 8.023575363333603365164842704085, 8.985935165219785889971174943893, 9.594286225156112457086858320355

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

\(L(\frac{3}{2})\)	\(\approx\)	\(1.176436764\)
\(L(\frac12)\)	\(\approx\)	\(1.176436764\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more