LMFDB - L-function 2-90e2-5.4-c1-0-30

Dirichlet series

L(s) = 1

− 0.0864i·7^-s + 0.913·11^-s + 2.62i·13^-s + 2.08i·17^-s − 4.93·19^-s − 8.47i·23^-s + 2.39·29^-s + 3.62·31^-s + 5.85i·37^-s − 6.64·41^-s − 8.24i·43^-s + 2.68i·47^-s + 6.99·49^-s + 5.73i·53^-s + 12.3·59^-s + ⋯

L(s) = 1

− 0.0326i·7^-s + 0.275·11^-s + 0.728i·13^-s + 0.506i·17^-s − 1.13·19^-s − 1.76i·23^-s + 0.445·29^-s + 0.651·31^-s + 0.962i·37^-s − 1.03·41^-s − 1.25i·43^-s + 0.392i·47^-s + 0.998·49^-s + 0.787i·53^-s + 1.60·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8100$ = $2^{2} \cdot 3^{4} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$64.6788$
Root analytic conductor:	$8.04231$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8100} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8100,\ (\ :1/2),\ 0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.821971593$
$L(\frac12)$	$\approx$	$1.821971593$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 0.0864iT - 7T^{2} $
	11	$ 1 - 0.913T + 11T^{2} $
	13	$ 1 - 2.62iT - 13T^{2} $
	17	$ 1 - 2.08iT - 17T^{2} $
	19	$ 1 + 4.93T + 19T^{2} $
	23	$ 1 + 8.47iT - 23T^{2} $
	29	$ 1 - 2.39T + 29T^{2} $
	31	$ 1 - 3.62T + 31T^{2} $
	37	$ 1 - 5.85iT - 37T^{2} $
	41	$ 1 + 6.64T + 41T^{2} $
	43	$ 1 + 8.24iT - 43T^{2} $
	47	$ 1 - 2.68iT - 47T^{2} $
	53	$ 1 - 5.73iT - 53T^{2} $
	59	$ 1 - 12.3T + 59T^{2} $
	61	$ 1 + 6.33T + 61T^{2} $
	67	$ 1 + 6.16iT - 67T^{2} $
	71	$ 1 - 12.3T + 71T^{2} $
	73	$ 1 - 5.31iT - 73T^{2} $
	79	$ 1 - 13.4T + 79T^{2} $
	83	$ 1 - 6.07iT - 83T^{2} $
	89	$ 1 + 8.13T + 89T^{2} $
	97	$ 1 + 11.1iT - 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

−8.081128730707212177043930088592, −6.90975739076113583933875162068, −6.62596491635412503117545990968, −5.95475732217475367625455256770, −4.93230833119049945947734471836, −4.34687489456415856487562112287, −3.69936042011946549955127218488, −2.60155688874957401103814421705, −1.93601251434315981526591298913, −0.75776949097187387591155256240, 0.58696420312106128655798756174, 1.70227411346612584715228358013, 2.62437178286639209039982698159, 3.47310418290975718616008135052, 4.16113057035459589422276546434, 5.08925496586377511130375452103, 5.61530449389149773368470751217, 6.46172357572007415983995697414, 7.03832625238680315570861566395, 7.86016523101154801789090335180

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

\(L(1)\)	\(\approx\)	\(1.821971593\)
\(L(\frac12)\)	\(\approx\)	\(1.821971593\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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