LMFDB - L-function 2-2800-5.4-c1-0-48

Dirichlet series

L(s) = 1

− i·7^-s + 3·9^-s − 4·11^-s − 6i·13^-s − 2i·17^-s − 6·29^-s − 8·31^-s + 10i·37^-s + 2·41^-s − 4i·43^-s + 8i·47^-s − 49^-s − 2i·53^-s − 8·59^-s − 14·61^-s + ⋯

L(s) = 1

− 0.377i·7^-s + 9^-s − 1.20·11^-s − 1.66i·13^-s − 0.485i·17^-s − 1.11·29^-s − 1.43·31^-s + 1.64i·37^-s + 0.312·41^-s − 0.609i·43^-s + 1.16i·47^-s − 0.142·49^-s − 0.274i·53^-s − 1.04·59^-s − 1.79·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2800$ = $2^{4} \cdot 5^{2} \cdot 7$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$22.3581$
Root analytic conductor:	$4.72843$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2800,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.7464900134$
$L(\frac12)$	$\approx$	$0.7464900134$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	7	$ 1 + iT $
good	3	$ 1 - 3T^{2} $
	11	$ 1 + 4T + 11T^{2} $
	13	$ 1 + 6iT - 13T^{2} $
	17	$ 1 + 2iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 + 6T + 29T^{2} $
	31	$ 1 + 8T + 31T^{2} $
	37	$ 1 - 10iT - 37T^{2} $
	41	$ 1 - 2T + 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 - 8iT - 47T^{2} $
	53	$ 1 + 2iT - 53T^{2} $
	59	$ 1 + 8T + 59T^{2} $
	61	$ 1 + 14T + 61T^{2} $
	67	$ 1 + 12iT - 67T^{2} $
	71	$ 1 - 16T + 71T^{2} $
	73	$ 1 - 2iT - 73T^{2} $
	79	$ 1 + 8T + 79T^{2} $
	83	$ 1 + 8iT - 83T^{2} $
	89	$ 1 + 10T + 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

−8.265899950052732115419759938554, −7.63705964918998725770461433039, −7.24858340210446861545299137651, −6.11208397826544489742084323069, −5.29690891153501965584474231139, −4.67549331470859371004058129327, −3.55905166563266995484188735299, −2.80724664657204643598495483787, −1.55284920240425145576299385101, −0.22725873497587523262646158805, 1.64769041325788342416736408368, 2.33132608666228447039110626194, 3.68288765754575919343428686690, 4.35358457679376603801551612536, 5.27684196099387196231529700968, 6.01725356963673534933512591456, 7.04534924918139999345931367787, 7.45560266832340850254893034282, 8.377609988003526667801267576654, 9.295547717678141579623653251891

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7464900134\)
\(L(\frac12)\)	\(\approx\)	\(0.7464900134\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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