LMFDB - L-function 2-70e2-5.4-c1-0-31

Dirichlet series

L(s) = 1

− 0.874i·3^-s + 2.23·9^-s + 3.47·11^-s + 2.28i·13^-s + 1.74i·17^-s − 0.333·19^-s − 5.47i·23^-s − 4.57i·27^-s + 4.23·29^-s + 1.20·31^-s − 3.03i·33^-s − 0.236i·37^-s + 2·39^-s − 1.95·41^-s − 8.23i·43^-s + ⋯

L(s) = 1

− 0.504i·3^-s + 0.745·9^-s + 1.04·11^-s + 0.634i·13^-s + 0.423i·17^-s − 0.0765·19^-s − 1.14i·23^-s − 0.880i·27^-s + 0.786·29^-s + 0.216·31^-s − 0.528i·33^-s − 0.0388i·37^-s + 0.320·39^-s − 0.305·41^-s − 1.25i·43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4900$ = $2^{2} \cdot 5^{2} \cdot 7^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$39.1266$
Root analytic conductor:	$6.25513$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4900} (2549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4900,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$2.336159842$
$L(\frac12)$	$\approx$	$2.336159842$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	7	$ 1 $
good	3	$ 1 + 0.874iT - 3T^{2} $
	11	$ 1 - 3.47T + 11T^{2} $
	13	$ 1 - 2.28iT - 13T^{2} $
	17	$ 1 - 1.74iT - 17T^{2} $
	19	$ 1 + 0.333T + 19T^{2} $
	23	$ 1 + 5.47iT - 23T^{2} $
	29	$ 1 - 4.23T + 29T^{2} $
	31	$ 1 - 1.20T + 31T^{2} $
	37	$ 1 + 0.236iT - 37T^{2} $
	41	$ 1 + 1.95T + 41T^{2} $
	43	$ 1 + 8.23iT - 43T^{2} $
	47	$ 1 - 7.73iT - 47T^{2} $
	53	$ 1 - 1.70iT - 53T^{2} $
	59	$ 1 + 5.11T + 59T^{2} $
	61	$ 1 - 14.6T + 61T^{2} $
	67	$ 1 - 3.94iT - 67T^{2} $
	71	$ 1 - 3.29T + 71T^{2} $
	73	$ 1 - 14.6iT - 73T^{2} $
	79	$ 1 - 2.52T + 79T^{2} $
	83	$ 1 - 9.02iT - 83T^{2} $
	89	$ 1 - 15.3T + 89T^{2} $
	97	$ 1 - 12.1iT - 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

−8.260601904903660189849910517678, −7.37144169257056711286363985467, −6.59959973810866465695968902188, −6.44751719150609068150797014970, −5.28594342465530179814517751615, −4.29518964524706713985119897225, −3.91856318035937396429895618043, −2.63560503673870686194434281057, −1.74133180656111840249329435105, −0.870399978129041686692866716495, 0.906374930775081013627623394138, 1.89801809607118125196645979346, 3.17736928476432911444484070313, 3.77253751565483877962123181501, 4.63234520784146947024844422401, 5.23718265130157534917579314670, 6.21273972549388719754154495407, 6.85406966430489700419297818664, 7.59162055040451576173163212831, 8.320278904821672093323068645591

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

\(L(1)\)	\(\approx\)	\(2.336159842\)
\(L(\frac12)\)	\(\approx\)	\(2.336159842\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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