LMFDB - L-function 2-90e2-5.4-c1-0-42

Dirichlet series

L(s) = 1

+ 4.99i·7^-s − 3.99·11^-s + 1.54i·13^-s − 6.99i·17^-s + 2.25·19^-s − 7.79i·23^-s + 6.16·29^-s − 0.543·31^-s + 6.25i·37^-s − 0.195·41^-s − 0.0863i·43^-s − 3.82i·47^-s − 17.9·49^-s − 4.19i·53^-s + 7.02·59^-s + ⋯

L(s) = 1

+ 1.88i·7^-s − 1.20·11^-s + 0.427i·13^-s − 1.69i·17^-s + 0.517·19^-s − 1.62i·23^-s + 1.14·29^-s − 0.0975·31^-s + 1.02i·37^-s − 0.0305·41^-s − 0.0131i·43^-s − 0.558i·47^-s − 2.56·49^-s − 0.575i·53^-s + 0.914·59^-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

$L(1)$	$\approx$	$1.440166594$
$L(\frac12)$	$\approx$	$1.440166594$
$L(\frac{3}{2})$		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 - 4.99iT - 7T^{2} $
	11	$ 1 + 3.99T + 11T^{2} $
	13	$ 1 - 1.54iT - 13T^{2} $
	17	$ 1 + 6.99iT - 17T^{2} $
	19	$ 1 - 2.25T + 19T^{2} $
	23	$ 1 + 7.79iT - 23T^{2} $
	29	$ 1 - 6.16T + 29T^{2} $
	31	$ 1 + 0.543T + 31T^{2} $
	37	$ 1 - 6.25iT - 37T^{2} $
	41	$ 1 + 0.195T + 41T^{2} $
	43	$ 1 + 0.0863iT - 43T^{2} $
	47	$ 1 + 3.82iT - 47T^{2} $
	53	$ 1 + 4.19iT - 53T^{2} $
	59	$ 1 - 7.02T + 59T^{2} $
	61	$ 1 + 2.90T + 61T^{2} $
	67	$ 1 + 8.96iT - 67T^{2} $
	71	$ 1 + 8.79T + 71T^{2} $
	73	$ 1 + 2.28iT - 73T^{2} $
	79	$ 1 - 12.6T + 79T^{2} $
	83	$ 1 - 13.9iT - 83T^{2} $
	89	$ 1 + 10.3T + 89T^{2} $
	97	$ 1 + 9.33iT - 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

−7.969881252773259398752794463432, −6.96669393420179422262424645200, −6.42437836565314813783960792340, −5.52200823634457256142223973851, −5.08673124225883337781183578261, −4.51017947027165484965704387576, −2.94271643901735542074949130032, −2.79115368253335229020475893623, −1.94800773358473241534289249285, −0.42661266810345945512234773613, 0.818081180176931306366617798483, 1.64722224696127381487706073064, 2.88543416800715975463109602307, 3.67684843420570549168037874669, 4.19143608452153431543119187471, 5.08895737741178063680149676511, 5.76504909158695825511441462249, 6.57074744650724622675287002495, 7.42759534290159414985767995047, 7.71208293220986808146310032348

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

\(L(1)\)	\(\approx\)	\(1.440166594\)
\(L(\frac12)\)	\(\approx\)	\(1.440166594\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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