LMFDB - L-function 2-4608-24.11-c1-0-9

Dirichlet series

L(s) = 1

− 3.69·5^-s + 3.41i·7^-s − 2.16i·11^-s − 2.16i·13^-s − 1.41i·17^-s − 7.39·19^-s + 4.82·23^-s + 8.65·25^-s + 6.75·29^-s − 2.24i·31^-s − 12.6i·35^-s + 7.39i·37^-s − 12.2i·41^-s − 10.4·43^-s − 3.17·47^-s + ⋯

L(s) = 1

− 1.65·5^-s + 1.29i·7^-s − 0.652i·11^-s − 0.600i·13^-s − 0.342i·17^-s − 1.69·19^-s + 1.00·23^-s + 1.73·25^-s + 1.25·29^-s − 0.402i·31^-s − 2.13i·35^-s + 1.21i·37^-s − 1.91i·41^-s − 1.59·43^-s − 0.462·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (2303, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ 0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.8861192028$
$L(\frac12)$	$\approx$	$0.8861192028$
$L(\frac{3}{2})$		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 + 3.69T + 5T^{2} $
	7	$ 1 - 3.41iT - 7T^{2} $
	11	$ 1 + 2.16iT - 11T^{2} $
	13	$ 1 + 2.16iT - 13T^{2} $
	17	$ 1 + 1.41iT - 17T^{2} $
	19	$ 1 + 7.39T + 19T^{2} $
	23	$ 1 - 4.82T + 23T^{2} $
	29	$ 1 - 6.75T + 29T^{2} $
	31	$ 1 + 2.24iT - 31T^{2} $
	37	$ 1 - 7.39iT - 37T^{2} $
	41	$ 1 + 12.2iT - 41T^{2} $
	43	$ 1 + 10.4T + 43T^{2} $
	47	$ 1 + 3.17T + 47T^{2} $
	53	$ 1 + 6.75T + 53T^{2} $
	59	$ 1 + 10.4iT - 59T^{2} $
	61	$ 1 + 3.06iT - 61T^{2} $
	67	$ 1 - 11.7T + 67T^{2} $
	71	$ 1 - 6.48T + 71T^{2} $
	73	$ 1 - 3.65T + 73T^{2} $
	79	$ 1 - 17.0iT - 79T^{2} $
	83	$ 1 - 8.28iT - 83T^{2} $
	89	$ 1 - 7.07iT - 89T^{2} $
	97	$ 1 + 11.3T + 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

−8.228185343232965247456850396496, −8.147879377027610519161613263272, −6.86066753227372583555923657034, −6.44772284270076457715000166934, −5.30699109779104685553170398780, −4.80040857919491034161474097141, −3.77946350430747166522881848639, −3.12763303822054493917142841911, −2.28392367484450526780399521688, −0.66861238225639759141956693853, 0.40605534050304296924723478286, 1.59849970750628766487469758507, 2.99823846052949708356775342081, 3.81260889120664694057037852204, 4.49063243097788521699030483748, 4.73957998498222161468582666911, 6.37226547211050303164905500912, 6.90789284835619635159806021140, 7.41893271954982948758765943313, 8.220123187082444506434365384714

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.8861192028\)
\(L(\frac12)\)	\(\approx\)	\(0.8861192028\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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