LMFDB - L-function 2-525-5.4-c5-0-56

Dirichlet series

L(s) = 1

− 3.77i·2^-s + 9i·3^-s + 17.7·4^-s + 33.9·6^-s − 49i·7^-s − 187. i·8^-s − 81·9^-s − 454.·11^-s + 159. i·12^-s + 174. i·13^-s − 184.·14^-s − 139.·16^-s + 1.56e3i·17^-s + 305. i·18^-s + 2.96e3·19^-s + ⋯

L(s) = 1

− 0.666i·2^-s + 0.577i·3^-s + 0.555·4^-s + 0.384·6^-s − 0.377i·7^-s − 1.03i·8^-s − 0.333·9^-s − 1.13·11^-s + 0.320i·12^-s + 0.286i·13^-s − 0.252·14^-s − 0.136·16^-s + 1.31i·17^-s + 0.222i·18^-s + 1.88·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$525$ = $3 \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$84.2015$
Root analytic conductor:	$9.17613$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{525} (274, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 525,\ (\ :5/2),\ 0.447 + 0.894i)$

Particular Values

$L(3)$	$\approx$	$2.414367108$
$L(\frac12)$	$\approx$	$2.414367108$
$L(\frac{7}{2})$		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 - 9iT $
	5	$ 1 $
	7	$ 1 + 49iT $
good	2	$ 1 + 3.77iT - 32T^{2} $
	11	$ 1 + 454.T + 1.61e5T^{2} $
	13	$ 1 - 174. iT - 3.71e5T^{2} $
	17	$ 1 - 1.56e3iT - 1.41e6T^{2} $
	19	$ 1 - 2.96e3T + 2.47e6T^{2} $
	23	$ 1 + 1.30e3iT - 6.43e6T^{2} $
	29	$ 1 + 1.98e3T + 2.05e7T^{2} $
	31	$ 1 - 2.44e3T + 2.86e7T^{2} $
	37	$ 1 + 1.81e3iT - 6.93e7T^{2} $
	41	$ 1 - 5.43e3T + 1.15e8T^{2} $
	43	$ 1 + 1.30e4iT - 1.47e8T^{2} $
	47	$ 1 - 1.12e4iT - 2.29e8T^{2} $
	53	$ 1 + 1.08e4iT - 4.18e8T^{2} $
	59	$ 1 + 6.47e3T + 7.14e8T^{2} $
	61	$ 1 - 4.53e4T + 8.44e8T^{2} $
	67	$ 1 + 4.13e4iT - 1.35e9T^{2} $
	71	$ 1 - 3.35e4T + 1.80e9T^{2} $
	73	$ 1 + 4.25e4iT - 2.07e9T^{2} $
	79	$ 1 - 2.60e4T + 3.07e9T^{2} $
	83	$ 1 + 1.88e4iT - 3.93e9T^{2} $
	89	$ 1 - 8.59e4T + 5.58e9T^{2} $
	97	$ 1 + 1.47e5iT - 8.58e9T^{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

−10.17204782468609175461826470477, −9.366364007653292953859676511649, −8.094005696791507825368378887149, −7.29550263717741491008392319346, −6.15359695222886264661087433861, −5.13099644733449735909112033693, −3.89880049292323450951139214417, −3.06688474560327608679372191743, −1.95403924727701598139891199761, −0.63677269173800617109057925336, 0.936208748329135500678718937013, 2.37920019456623223308198373389, 3.09971155855384606107095728547, 5.16598362991234763081161893850, 5.55106211309470739529143962685, 6.73766243346380308379106148156, 7.56415507110813759583865995696, 7.986779754771630698564748171933, 9.211257046839535926802212161272, 10.18613389594976023192327614659

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(2.414367108\)
\(L(\frac12)\)	\(\approx\)	\(2.414367108\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more