LMFDB - L-function 2-90e2-5.4-c1-0-18

Dirichlet series

L(s) = 1

− 4i·7^-s + 3·11^-s + 4i·13^-s − 5·19^-s + 6i·23^-s + 9·29^-s + 5·31^-s + 2i·37^-s − 9·41^-s + 10i·43^-s − 6i·47^-s − 9·49^-s + 12i·53^-s − 9·59^-s − 10·61^-s + ⋯

L(s) = 1

− 1.51i·7^-s + 0.904·11^-s + 1.10i·13^-s − 1.14·19^-s + 1.25i·23^-s + 1.67·29^-s + 0.898·31^-s + 0.328i·37^-s − 1.40·41^-s + 1.52i·43^-s − 0.875i·47^-s − 1.28·49^-s + 1.64i·53^-s − 1.17·59^-s − 1.28·61^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$8100$ = $2^{2} \cdot 3^{4} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
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:	$(2,\ 8100,\ (\ :1/2),\ 0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.580075607$
$L(\frac12)$	$\approx$	$1.580075607$
$L(\frac{3}{2})$		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 + 4iT - 7T^{2} $
	11	$ 1 - 3T + 11T^{2} $
	13	$ 1 - 4iT - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 5T + 19T^{2} $
	23	$ 1 - 6iT - 23T^{2} $
	29	$ 1 - 9T + 29T^{2} $
	31	$ 1 - 5T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 + 9T + 41T^{2} $
	43	$ 1 - 10iT - 43T^{2} $
	47	$ 1 + 6iT - 47T^{2} $
	53	$ 1 - 12iT - 53T^{2} $
	59	$ 1 + 9T + 59T^{2} $
	61	$ 1 + 10T + 61T^{2} $
	67	$ 1 - 2iT - 67T^{2} $
	71	$ 1 - 3T + 71T^{2} $
	73	$ 1 - 4iT - 73T^{2} $
	79	$ 1 - 4T + 79T^{2} $
	83	$ 1 + 6iT - 83T^{2} $
	89	$ 1 - 9T + 89T^{2} $
	97	$ 1 - 2iT - 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

−7.896991764601046083267678194026, −7.16367256370461133833970837946, −6.53239127785860156602342407887, −6.23906877904912093020757912163, −4.85091576748896371807942654703, −4.38374202551815685015196066995, −3.81109860777580603617551319637, −2.94407829053737253396460231590, −1.67560496926417688775667522351, −1.05487103233014165842616105893, 0.39933647876750581816733663898, 1.71680167446038212206007107819, 2.57730295356432369557971685450, 3.16774684067533091896441644117, 4.24118536048522755757380064591, 4.95059647334067404023635441222, 5.64383531484979410889158488991, 6.51472452494962458507393716037, 6.60597526108958956137347048781, 8.014314369223243594461048982039

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.580075607\)
\(L(\frac12)\)	\(\approx\)	\(1.580075607\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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