LMFDB - L-function 2-5328-12.11-c1-0-60

Dirichlet series

L(s) = 1

− 1.00i·5^-s − 3.02i·7^-s − 0.572·13^-s − 8.05i·17^-s + 7.53i·19^-s + 2.61·23^-s + 3.98·25^-s + 6.03i·29^-s − 5.18i·31^-s − 3.05·35^-s + 37^-s − 4.24i·41^-s − 9.88i·43^-s + 7.60·47^-s − 2.14·49^-s + ⋯

L(s) = 1

− 0.451i·5^-s − 1.14i·7^-s − 0.158·13^-s − 1.95i·17^-s + 1.72i·19^-s + 0.545·23^-s + 0.796·25^-s + 1.12i·29^-s − 0.930i·31^-s − 0.515·35^-s + 0.164·37^-s − 0.662i·41^-s − 1.50i·43^-s + 1.10·47^-s − 0.306·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5328$ = $2^{4} \cdot 3^{2} \cdot 37$
Sign:	$-0.577 + 0.816i$
Analytic conductor:	$42.5442$
Root analytic conductor:	$6.52259$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5328} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5328,\ (\ :1/2),\ -0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.579196573$
$L(\frac12)$	$\approx$	$1.579196573$
$L(\frac{3}{2})$		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 $
	37	$ 1 - T $
good	5	$ 1 + 1.00iT - 5T^{2} $
	7	$ 1 + 3.02iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 0.572T + 13T^{2} $
	17	$ 1 + 8.05iT - 17T^{2} $
	19	$ 1 - 7.53iT - 19T^{2} $
	23	$ 1 - 2.61T + 23T^{2} $
	29	$ 1 - 6.03iT - 29T^{2} $
	31	$ 1 + 5.18iT - 31T^{2} $
	41	$ 1 + 4.24iT - 41T^{2} $
	43	$ 1 + 9.88iT - 43T^{2} $
	47	$ 1 - 7.60T + 47T^{2} $
	53	$ 1 + 6.26iT - 53T^{2} $
	59	$ 1 - 5.66T + 59T^{2} $
	61	$ 1 + 7.96T + 61T^{2} $
	67	$ 1 - 7.33iT - 67T^{2} $
	71	$ 1 - 13.7T + 71T^{2} $
	73	$ 1 + 3.01T + 73T^{2} $
	79	$ 1 + 13.5iT - 79T^{2} $
	83	$ 1 + 7.60T + 83T^{2} $
	89	$ 1 + 7.47iT - 89T^{2} $
	97	$ 1 + 10.5T + 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

−7.78156994692362069111334097443, −7.24630237270471689314592081163, −6.74003799665666980006442995892, −5.57713996514117257256526015088, −5.07068519629956069273096011386, −4.19404205296727877237316271903, −3.54009711937682698058584928787, −2.53964497911815486537808835060, −1.32385362356273293164300133710, −0.45358561626883433951240143409, 1.24682161100322985902575197849, 2.46781135926594348469033253767, 2.88079303558850862137038516908, 4.00803853641148194702134769003, 4.83203156261530433155978703807, 5.60084905812631349875309767761, 6.36655566833014245436913834719, 6.81749944499726600875498250893, 7.82951110822713294399868702134, 8.466847260986924738732499250212

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.579196573\)
\(L(\frac12)\)	\(\approx\)	\(1.579196573\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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