LMFDB - L-function 2-1127-161.160-c1-0-62

Dirichlet series

L(s) = 1

+ 2.54·2^-s − 0.893i·3^-s + 4.49·4^-s − 2.27i·6^-s + 6.36·8^-s + 2.20·9^-s − 4.02i·12^-s + 2.48i·13^-s + 7.23·16^-s + 5.61·18^-s + 4.79·23^-s − 5.69i·24^-s − 5·25^-s + 6.32i·26^-s − 4.64i·27^-s + ⋯

L(s) = 1

+ 1.80·2^-s − 0.516i·3^-s + 2.24·4^-s − 0.930i·6^-s + 2.25·8^-s + 0.733·9^-s − 1.16i·12^-s + 0.688i·13^-s + 1.80·16^-s + 1.32·18^-s + 1.00·23^-s − 1.16i·24^-s − 25^-s + 1.24i·26^-s − 0.894i·27^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1127$ = $7^{2} \cdot 23$
Sign:	$0.912 + 0.409i$
Analytic conductor:	$8.99914$
Root analytic conductor:	$2.99985$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1127} (1126, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1127,\ (\ :1/2),\ 0.912 + 0.409i)$

Particular Values

$L(1)$	$\approx$	$5.012089921$
$L(\frac12)$	$\approx$	$5.012089921$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	7	$ 1 $
	23	$ 1 - 4.79T $
good	2	$ 1 - 2.54T + 2T^{2} $
	3	$ 1 + 0.893iT - 3T^{2} $
	5	$ 1 + 5T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 - 2.48iT - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 + 19T^{2} $
	29	$ 1 + 6.70T + 29T^{2} $
	31	$ 1 + 4.54iT - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 + 12.7iT - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 - 10.6iT - 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 - 14.7iT - 59T^{2} $
	61	$ 1 + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 14.0T + 71T^{2} $
	73	$ 1 - 17.0iT - 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 + 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

−9.973345816302181882365291165928, −8.934562156413334514007846932655, −7.45871460866168554735379852737, −7.19962773347397530538165154963, −6.17894013779057622726054391222, −5.51345892862312186846599405438, −4.40015312751536009286051866415, −3.85615029850207714527325454267, −2.58501813721742578294643022206, −1.62042927261895701082694012344, 1.75897450465415513833261066341, 3.10811893878716614086536469966, 3.77014047699622932868807005737, 4.73780615004010974241403511480, 5.30151817176759316305364423517, 6.25972177178299932695338590010, 7.09711631998868157309926536954, 7.898089952528120247314598844739, 9.243693549247054813164598737039, 10.16318742648892212340432778693

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(5.012089921\)
\(L(\frac12)\)	\(\approx\)	\(5.012089921\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more