LMFDB - L-function 2-90e2-5.4-c1-0-44

Dirichlet series

L(s) = 1

− 3.40i·7^-s + 4.40·11^-s − 2.06i·13^-s + 1.40i·17^-s + 6.35·19^-s + 0.107i·23^-s + 9.08·29^-s + 3.06·31^-s + 1.95i·37^-s + 8.69·41^-s + 7.12i·43^-s + 7.48i·47^-s − 4.57·49^-s + 13.0i·53^-s − 13.1·59^-s + ⋯

L(s) = 1

− 1.28i·7^-s + 1.32·11^-s − 0.572i·13^-s + 0.339i·17^-s + 1.45·19^-s + 0.0224i·23^-s + 1.68·29^-s + 0.550·31^-s + 0.321i·37^-s + 1.35·41^-s + 1.08i·43^-s + 1.09i·47^-s − 0.653·49^-s + 1.79i·53^-s − 1.71·59^-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

$L(1)$	$\approx$	$2.606594236$
$L(\frac12)$	$\approx$	$2.606594236$
$L(\frac{3}{2})$		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 + 3.40iT - 7T^{2} $
	11	$ 1 - 4.40T + 11T^{2} $
	13	$ 1 + 2.06iT - 13T^{2} $
	17	$ 1 - 1.40iT - 17T^{2} $
	19	$ 1 - 6.35T + 19T^{2} $
	23	$ 1 - 0.107iT - 23T^{2} $
	29	$ 1 - 9.08T + 29T^{2} $
	31	$ 1 - 3.06T + 31T^{2} $
	37	$ 1 - 1.95iT - 37T^{2} $
	41	$ 1 - 8.69T + 41T^{2} $
	43	$ 1 - 7.12iT - 43T^{2} $
	47	$ 1 - 7.48iT - 47T^{2} $
	53	$ 1 - 13.0iT - 53T^{2} $
	59	$ 1 + 13.1T + 59T^{2} $
	61	$ 1 + 1.72T + 61T^{2} $
	67	$ 1 + 12.3iT - 67T^{2} $
	71	$ 1 - 7.50T + 71T^{2} $
	73	$ 1 - 5.42iT - 73T^{2} $
	79	$ 1 + 9.43T + 79T^{2} $
	83	$ 1 - 8.97iT - 83T^{2} $
	89	$ 1 - 4.01T + 89T^{2} $
	97	$ 1 - 2.17iT - 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

−7.70509171485870550774726639127, −7.15896086391879263935503354743, −6.37348502290472920082750626122, −5.90175850390141875498012949431, −4.68189186416327281733174296902, −4.35326656571388726783817239699, −3.42488743901711803047754925973, −2.82780100115736097518705949044, −1.27499455941255742212211371943, −0.941852022986398998880713860886, 0.864591370463115597088652923394, 1.87144877059322132331398486824, 2.72587274536652532875653306141, 3.49511141784964856650706418865, 4.39026875366908581021543223330, 5.12887135432301728485942309040, 5.82445381223636642791123421621, 6.49715795611655815401899117845, 7.06644590890205882420697622414, 7.919077639938966810451893859547

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

\(L(1)\)	\(\approx\)	\(2.606594236\)
\(L(\frac12)\)	\(\approx\)	\(2.606594236\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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