LMFDB - L-function 2-1127-161.160-c1-0-65

Dirichlet series

L(s) = 1

− 2.54·2^-s − 3.34i·3^-s + 4.49·4^-s + 8.53i·6^-s − 6.36·8^-s − 8.20·9^-s − 15.0i·12^-s − 6.77i·13^-s + 7.23·16^-s + 20.9·18^-s + 4.79·23^-s + 21.3i·24^-s − 5·25^-s + 17.2i·26^-s + 17.4i·27^-s + ⋯

L(s) = 1

− 1.80·2^-s − 1.93i·3^-s + 2.24·4^-s + 3.48i·6^-s − 2.25·8^-s − 2.73·9^-s − 4.34i·12^-s − 1.87i·13^-s + 1.80·16^-s + 4.92·18^-s + 1.00·23^-s + 4.35i·24^-s − 25^-s + 3.38i·26^-s + 3.34i·27^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1127$ = $7^{2} \cdot 23$
Sign:	$-0.156 - 0.987i$
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:	$(2,\ 1127,\ (\ :1/2),\ -0.156 - 0.987i)$

Particular Values

$L(1)$	$\approx$	$0.2510334883$
$L(\frac12)$	$\approx$	$0.2510334883$
$L(\frac{3}{2})$		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 + 3.34iT - 3T^{2} $
	5	$ 1 + 5T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 6.77iT - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 + 19T^{2} $
	29	$ 1 + 6.70T + 29T^{2} $
	31	$ 1 + 10.1iT - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 0.987iT - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 - 8.61iT - 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 - 4.26iT - 59T^{2} $
	61	$ 1 + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 14.0T + 71T^{2} $
	73	$ 1 - 1.17iT - 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.990749632574807053057190339079, −8.191206582225484632439265772284, −7.62537264241238050497832295304, −7.30230901421023206290924431347, −6.15470831196934065236023506876, −5.66957570581116654291420145557, −3.06906820930652488472601553333, −2.23488961508807332227161259283, −1.15468526990917564733860842361, −0.21556333019660758790878453055, 1.91991739721287333490679969570, 3.22294624526648296090049032042, 4.24759569664494539459509241107, 5.31377875159097420134897193662, 6.43849482232625393709164669968, 7.34015648628121494422311406722, 8.575275729840034868361088559003, 8.956888360528917366720892846612, 9.536248482732909854396453187064, 10.14195159101239665595191089962

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2510334883\)
\(L(\frac12)\)	\(\approx\)	\(0.2510334883\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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