LMFDB - L-function 2-4608-8.5-c1-0-29

Dirichlet series

L(s) = 1

− 2.44i·5^-s − 1.41·7^-s + 3.46i·11^-s − 4.89i·13^-s − 4·17^-s + 6.92i·19^-s + 5.65·23^-s − 0.999·25^-s + 2.44i·29^-s − 1.41·31^-s + 3.46i·35^-s + 4.89i·37^-s + 4·41^-s + 6.92i·43^-s + 5.65·47^-s + ⋯

L(s) = 1

− 1.09i·5^-s − 0.534·7^-s + 1.04i·11^-s − 1.35i·13^-s − 0.970·17^-s + 1.58i·19^-s + 1.17·23^-s − 0.199·25^-s + 0.454i·29^-s − 0.254·31^-s + 0.585i·35^-s + 0.805i·37^-s + 0.624·41^-s + 1.05i·43^-s + 0.825·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$1$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (2305, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$1.566820431$
$L(\frac12)$	$\approx$	$1.566820431$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 2.44iT - 5T^{2} $
	7	$ 1 + 1.41T + 7T^{2} $
	11	$ 1 - 3.46iT - 11T^{2} $
	13	$ 1 + 4.89iT - 13T^{2} $
	17	$ 1 + 4T + 17T^{2} $
	19	$ 1 - 6.92iT - 19T^{2} $
	23	$ 1 - 5.65T + 23T^{2} $
	29	$ 1 - 2.44iT - 29T^{2} $
	31	$ 1 + 1.41T + 31T^{2} $
	37	$ 1 - 4.89iT - 37T^{2} $
	41	$ 1 - 4T + 41T^{2} $
	43	$ 1 - 6.92iT - 43T^{2} $
	47	$ 1 - 5.65T + 47T^{2} $
	53	$ 1 - 7.34iT - 53T^{2} $
	59	$ 1 + 13.8iT - 59T^{2} $
	61	$ 1 + 4.89iT - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 - 11.3T + 71T^{2} $
	73	$ 1 - 4T + 73T^{2} $
	79	$ 1 - 7.07T + 79T^{2} $
	83	$ 1 + 10.3iT - 83T^{2} $
	89	$ 1 - 16T + 89T^{2} $
	97	$ 1 - 6T + 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

−8.162372290880764417930078568726, −7.83700246621454478655768606995, −6.82327391010889795761457728611, −6.13642188448470139556361373195, −5.18361034246986985878454191459, −4.78404978067091110174540080179, −3.80672529059055861269468505735, −2.94040842756986581388114834508, −1.80578337397599410368536239306, −0.795153859598788601995396417518, 0.59865804161156221824948212926, 2.22343176331040973772526204922, 2.82733509435558492039644665651, 3.68637881722356267211728714117, 4.48801792416474167106171053078, 5.47599573282456919424688631099, 6.39775290930053493544652348249, 6.86648605787934016977000643031, 7.24654232333502098637902958770, 8.475879297258476418449259735167

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.566820431\)
\(L(\frac12)\)	\(\approx\)	\(1.566820431\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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