LMFDB - L-function 2-84e2-28.27-c1-0-31

Dirichlet series

L(s) = 1

− 1.73i·5^-s + 1.73i·11^-s + 3.46i·17^-s − 2·19^-s + 2.00·25^-s − 9·29^-s + 5·31^-s + 10·37^-s − 10.3i·41^-s − 3.46i·43^-s − 12·47^-s + 9·53^-s + 2.99·55^-s + 9·59^-s + 13.8i·67^-s + ⋯

L(s) = 1

− 0.774i·5^-s + 0.522i·11^-s + 0.840i·17^-s − 0.458·19^-s + 0.400·25^-s − 1.67·29^-s + 0.898·31^-s + 1.64·37^-s − 1.62i·41^-s − 0.528i·43^-s − 1.75·47^-s + 1.23·53^-s + 0.404·55^-s + 1.17·59^-s + 1.69i·67^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.944 - 0.327i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ 0.944 - 0.327i)$

Particular Values

$L(1)$	$\approx$	$1.754850734$
$L(\frac12)$	$\approx$	$1.754850734$
$L(\frac{3}{2})$		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 $
	7	$ 1 $
good	5	$ 1 + 1.73iT - 5T^{2} $
	11	$ 1 - 1.73iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 3.46iT - 17T^{2} $
	19	$ 1 + 2T + 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 + 9T + 29T^{2} $
	31	$ 1 - 5T + 31T^{2} $
	37	$ 1 - 10T + 37T^{2} $
	41	$ 1 + 10.3iT - 41T^{2} $
	43	$ 1 + 3.46iT - 43T^{2} $
	47	$ 1 + 12T + 47T^{2} $
	53	$ 1 - 9T + 53T^{2} $
	59	$ 1 - 9T + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 13.8iT - 67T^{2} $
	71	$ 1 - 13.8iT - 71T^{2} $
	73	$ 1 - 6.92iT - 73T^{2} $
	79	$ 1 - 5.19iT - 79T^{2} $
	83	$ 1 + 3T + 83T^{2} $
	89	$ 1 - 3.46iT - 89T^{2} $
	97	$ 1 - 19.0iT - 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

−8.167720925401891046948706943258, −7.23328218676107066828085211221, −6.67190983449897155422949657068, −5.69289053324231869194313502211, −5.26864435459295427135276181266, −4.23348982272405747351825015179, −3.91402477584844330865456522900, −2.63806026196075987504731752008, −1.82481290133913226245253154788, −0.816983947327139113656875989110, 0.55657533732187914545903509249, 1.83920404413216069526400203015, 2.84277555227717277885340193756, 3.30968187324325453840416597687, 4.35256486651412046606543450964, 5.01991687078121987870403959996, 6.00949109911626058410884031094, 6.43784893011807652739358521736, 7.21004650151500801986709774532, 7.86137978480158995060689498092

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.754850734\)
\(L(\frac12)\)	\(\approx\)	\(1.754850734\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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