LMFDB - L-function 2-4608-24.11-c1-0-38

Dirichlet series

L(s) = 1

− 1.53·5^-s − 0.585i·7^-s + 5.22i·11^-s + 5.22i·13^-s − 1.41i·17^-s − 3.06·19^-s − 0.828·23^-s − 2.65·25^-s − 5.86·29^-s − 6.24i·31^-s + 0.896i·35^-s − 3.06i·37^-s + 3.75i·41^-s + 4.32·43^-s − 8.82·47^-s + ⋯

L(s) = 1

− 0.684·5^-s − 0.221i·7^-s + 1.57i·11^-s + 1.44i·13^-s − 0.342i·17^-s − 0.702·19^-s − 0.172·23^-s − 0.531·25^-s − 1.08·29^-s − 1.12i·31^-s + 0.151i·35^-s − 0.503i·37^-s + 0.586i·41^-s + 0.660·43^-s − 1.28·47^-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

$L(1)$	$\approx$	$0.2230222004$
$L(\frac12)$	$\approx$	$0.2230222004$
$L(\frac{3}{2})$		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 + 1.53T + 5T^{2} $
	7	$ 1 + 0.585iT - 7T^{2} $
	11	$ 1 - 5.22iT - 11T^{2} $
	13	$ 1 - 5.22iT - 13T^{2} $
	17	$ 1 + 1.41iT - 17T^{2} $
	19	$ 1 + 3.06T + 19T^{2} $
	23	$ 1 + 0.828T + 23T^{2} $
	29	$ 1 + 5.86T + 29T^{2} $
	31	$ 1 + 6.24iT - 31T^{2} $
	37	$ 1 + 3.06iT - 37T^{2} $
	41	$ 1 - 3.75iT - 41T^{2} $
	43	$ 1 - 4.32T + 43T^{2} $
	47	$ 1 + 8.82T + 47T^{2} $
	53	$ 1 - 5.86T + 53T^{2} $
	59	$ 1 + 4.32iT - 59T^{2} $
	61	$ 1 + 7.39iT - 61T^{2} $
	67	$ 1 - 13.5T + 67T^{2} $
	71	$ 1 + 10.4T + 71T^{2} $
	73	$ 1 + 7.65T + 73T^{2} $
	79	$ 1 + 2.92iT - 79T^{2} $
	83	$ 1 - 9.55iT - 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

−7.905174193550910954542991716325, −7.32576587720803607484447781294, −6.81969028244980689605400717210, −5.96115189349710794153641894476, −4.88060232086837637760179269003, −4.21389774279863200097987256511, −3.80895473171603167709813689136, −2.36044586121742563949562274375, −1.73602605631198394780791844237, −0.06916462486139464884913268747, 1.00229815699266280318978667219, 2.41706462211640068492784210274, 3.38707805422878720317940105659, 3.80109112107341680134787409732, 4.95651883928625871576078795877, 5.76698122611046492809128804820, 6.17455881882520843323919997997, 7.33071710500886751030450820763, 7.87032524776353061629141232273, 8.637764699825355491378695289334

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

\(L(1)\)	\(\approx\)	\(0.2230222004\)
\(L(\frac12)\)	\(\approx\)	\(0.2230222004\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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