LMFDB - L-function 2-4032-28.27-c1-0-47

Dirichlet series

L(s) = 1

− 2.61i·5^-s + (−2.61 + 0.414i)7^-s − 2i·11^-s + 4.77i·13^-s − 3.06i·17^-s + 4.14·19^-s + 7.65i·23^-s − 1.82·25^-s + 3.65·29^-s + 3.06·31^-s + (1.08 + 6.82i)35^-s + 7.65·37^-s − 9.55i·41^-s − 3.65i·43^-s − 7.39·47^-s + ⋯

L(s) = 1

− 1.16i·5^-s + (−0.987 + 0.156i)7^-s − 0.603i·11^-s + 1.32i·13^-s − 0.742i·17^-s + 0.950·19^-s + 1.59i·23^-s − 0.365·25^-s + 0.679·29^-s + 0.549·31^-s + (0.182 + 1.15i)35^-s + 1.25·37^-s − 1.49i·41^-s − 0.557i·43^-s − 1.07·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.156 + 0.987i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3583, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.156 + 0.987i)$

Particular Values

$L(1)$	$\approx$	$1.483274452$
$L(\frac12)$	$\approx$	$1.483274452$
$L(\frac{3}{2})$		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 + (2.61 - 0.414i)T $
good	5	$ 1 + 2.61iT - 5T^{2} $
	11	$ 1 + 2iT - 11T^{2} $
	13	$ 1 - 4.77iT - 13T^{2} $
	17	$ 1 + 3.06iT - 17T^{2} $
	19	$ 1 - 4.14T + 19T^{2} $
	23	$ 1 - 7.65iT - 23T^{2} $
	29	$ 1 - 3.65T + 29T^{2} $
	31	$ 1 - 3.06T + 31T^{2} $
	37	$ 1 - 7.65T + 37T^{2} $
	41	$ 1 + 9.55iT - 41T^{2} $
	43	$ 1 + 3.65iT - 43T^{2} $
	47	$ 1 + 7.39T + 47T^{2} $
	53	$ 1 + 2T + 53T^{2} $
	59	$ 1 - 8.47T + 59T^{2} $
	61	$ 1 + 2.61iT - 61T^{2} $
	67	$ 1 - 15.6iT - 67T^{2} $
	71	$ 1 + 8.82iT - 71T^{2} $
	73	$ 1 + 12.6iT - 73T^{2} $
	79	$ 1 + 12.8iT - 79T^{2} $
	83	$ 1 + 11.5T + 83T^{2} $
	89	$ 1 - 2.16iT - 89T^{2} $
	97	$ 1 + 13.5iT - 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

−8.479084061972747912279654769354, −7.49187320423776679644622386613, −6.85085150730252899905860291745, −5.96076679497236115840906829929, −5.32458902588352390530203266638, −4.52958469482472445361121480325, −3.66744423522454597249413832134, −2.82703216514415358211034722334, −1.57369232916943217919340366304, −0.53312484116502612299258074107, 0.959768828220903411829932930217, 2.67290882222862017130548317206, 2.90366040021137227855991874528, 3.88320671383187596002585908261, 4.82802777713938889896251815630, 5.87666648161798336109407970459, 6.51416754905775326968063878929, 6.95647822765587905518763864301, 7.907969652178949365004334796315, 8.377751376143033233088410236499

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.483274452\)
\(L(\frac12)\)	\(\approx\)	\(1.483274452\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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