LMFDB - L-function 2-4032-28.27-c1-0-16

Dirichlet series

L(s) = 1

− 0.922i·5^-s + (−2.06 − 1.65i)7^-s − 1.61i·11^-s + 2.82i·13^-s + 0.922i·17^-s + 0.540·19^-s + 7.05i·23^-s + 4.14·25^-s + 1.01·29^-s − 8.67·31^-s + (−1.52 + 1.90i)35^-s − 8.19·37^-s + 6.57i·41^-s − 4.96i·43^-s + 2.60·47^-s + ⋯

L(s) = 1

− 0.412i·5^-s + (−0.781 − 0.624i)7^-s − 0.487i·11^-s + 0.784i·13^-s + 0.223i·17^-s + 0.123·19^-s + 1.47i·23^-s + 0.829·25^-s + 0.188·29^-s − 1.55·31^-s + (−0.257 + 0.322i)35^-s − 1.34·37^-s + 1.02i·41^-s − 0.756i·43^-s + 0.380·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.624 - 0.781i$
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:	$(2,\ 4032,\ (\ :1/2),\ 0.624 - 0.781i)$

Particular Values

$L(1)$	$\approx$	$1.217779454$
$L(\frac12)$	$\approx$	$1.217779454$
$L(\frac{3}{2})$		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 + (2.06 + 1.65i)T $
good	5	$ 1 + 0.922iT - 5T^{2} $
	11	$ 1 + 1.61iT - 11T^{2} $
	13	$ 1 - 2.82iT - 13T^{2} $
	17	$ 1 - 0.922iT - 17T^{2} $
	19	$ 1 - 0.540T + 19T^{2} $
	23	$ 1 - 7.05iT - 23T^{2} $
	29	$ 1 - 1.01T + 29T^{2} $
	31	$ 1 + 8.67T + 31T^{2} $
	37	$ 1 + 8.19T + 37T^{2} $
	41	$ 1 - 6.57iT - 41T^{2} $
	43	$ 1 + 4.96iT - 43T^{2} $
	47	$ 1 - 2.60T + 47T^{2} $
	53	$ 1 - 5.59T + 53T^{2} $
	59	$ 1 + 1.65T + 59T^{2} $
	61	$ 1 + 7.01iT - 61T^{2} $
	67	$ 1 + 6.80iT - 67T^{2} $
	71	$ 1 - 5.21iT - 71T^{2} $
	73	$ 1 - 7.50iT - 73T^{2} $
	79	$ 1 + 7.11iT - 79T^{2} $
	83	$ 1 - 17.9T + 83T^{2} $
	89	$ 1 - 17.9iT - 89T^{2} $
	97	$ 1 - 4.18iT - 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

−8.710823080441123084907934585160, −7.71745818804460058170472478049, −7.07685833131037036030407039215, −6.42474046577384226522267079773, −5.55657290750834906481024862964, −4.83377755036595062361580032519, −3.75319884736727180343483083290, −3.37415717735353594009020786304, −2.01609477966049875933764726286, −0.946390183662666333045840668658, 0.41518580245653130349598450888, 2.01313024033400201475429234170, 2.86053297741293849446757638245, 3.52537061795924437968025529531, 4.61561698109996376590288668162, 5.44987346535361807820359319945, 6.11867039109068159426074342131, 6.97029332077099646228511994282, 7.38685428250676619949625220547, 8.559269923537335682663532154882

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.217779454\)
\(L(\frac12)\)	\(\approx\)	\(1.217779454\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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