LMFDB - L-function 2-84e2-12.11-c1-0-5

Dirichlet series

L(s) = 1

− 0.732i·5^-s − 4.24·11^-s − 3.48·13^-s − 2.73i·17^-s − 1.79i·19^-s − 0.656·23^-s + 4.46·25^-s − 2.44i·29^-s − 6.69i·31^-s + 3.46·37^-s + 6.19i·41^-s − 2.53i·43^-s − 3.46·47^-s + 1.41i·53^-s + 3.10i·55^-s + ⋯

L(s) = 1

− 0.327i·5^-s − 1.27·11^-s − 0.966·13^-s − 0.662i·17^-s − 0.411i·19^-s − 0.136·23^-s + 0.892·25^-s − 0.454i·29^-s − 1.20i·31^-s + 0.569·37^-s + 0.967i·41^-s − 0.386i·43^-s − 0.505·47^-s + 0.194i·53^-s + 0.418i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.418 - 0.908i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (4607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.418 - 0.908i)$

Particular Values

$L(1)$	$\approx$	$0.4150980792$
$L(\frac12)$	$\approx$	$0.4150980792$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 + 0.732iT - 5T^{2} $
	11	$ 1 + 4.24T + 11T^{2} $
	13	$ 1 + 3.48T + 13T^{2} $
	17	$ 1 + 2.73iT - 17T^{2} $
	19	$ 1 + 1.79iT - 19T^{2} $
	23	$ 1 + 0.656T + 23T^{2} $
	29	$ 1 + 2.44iT - 29T^{2} $
	31	$ 1 + 6.69iT - 31T^{2} $
	37	$ 1 - 3.46T + 37T^{2} $
	41	$ 1 - 6.19iT - 41T^{2} $
	43	$ 1 + 2.53iT - 43T^{2} $
	47	$ 1 + 3.46T + 47T^{2} $
	53	$ 1 - 1.41iT - 53T^{2} $
	59	$ 1 - 10.3T + 59T^{2} $
	61	$ 1 - 3.20T + 61T^{2} $
	67	$ 1 - 0.928iT - 67T^{2} $
	71	$ 1 + 4.24T + 71T^{2} $
	73	$ 1 + 3.20T + 73T^{2} $
	79	$ 1 - 15.4iT - 79T^{2} $
	83	$ 1 + 9.46T + 83T^{2} $
	89	$ 1 - 7.66iT - 89T^{2} $
	97	$ 1 - 10.1T + 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

−8.086153417906748495330175692896, −7.51650823590071970908459483520, −6.89479873260203015229735749788, −5.99137622098751435723883872349, −5.16026242296449010889631618372, −4.81470913405242638995144684826, −3.90320648438163809019089200907, −2.68667243563036526387255894877, −2.42164225960277285780985688456, −0.935548871579024577715958377083, 0.11365654303248221709201379876, 1.56892806406112442696347586336, 2.56167518332529566576323646671, 3.13827856155495983722693542701, 4.13207312772123633681341113222, 5.01838072755963378909846228339, 5.46340997903680527943602196139, 6.39087909629874847011196514982, 7.09837508539968344005716826438, 7.66575819419332686773212828867

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4150980792\)
\(L(\frac12)\)	\(\approx\)	\(0.4150980792\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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