LMFDB - L-function 2-1152-12.11-c1-0-3

Dirichlet series

L(s) = 1

− 3.41i·5^-s + 4.82i·7^-s − 2.82·11^-s + 2.82·13^-s + 5.41i·17^-s + 5.65i·19^-s + 1.17·23^-s − 6.65·25^-s − 0.585i·29^-s − 3.17i·31^-s + 16.4·35^-s + 3.65·37^-s + 2.58i·41^-s + 9.65i·43^-s + 12.4·47^-s + ⋯

L(s) = 1

− 1.52i·5^-s + 1.82i·7^-s − 0.852·11^-s + 0.784·13^-s + 1.31i·17^-s + 1.29i·19^-s + 0.244·23^-s − 1.33·25^-s − 0.108i·29^-s − 0.569i·31^-s + 2.78·35^-s + 0.601·37^-s + 0.403i·41^-s + 1.47i·43^-s + 1.82·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$9.19876$
Root analytic conductor:	$3.03294$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1/2),\ 0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.394543776$
$L(\frac12)$	$\approx$	$1.394543776$
$L(\frac{3}{2})$		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 + 3.41iT - 5T^{2} $
	7	$ 1 - 4.82iT - 7T^{2} $
	11	$ 1 + 2.82T + 11T^{2} $
	13	$ 1 - 2.82T + 13T^{2} $
	17	$ 1 - 5.41iT - 17T^{2} $
	19	$ 1 - 5.65iT - 19T^{2} $
	23	$ 1 - 1.17T + 23T^{2} $
	29	$ 1 + 0.585iT - 29T^{2} $
	31	$ 1 + 3.17iT - 31T^{2} $
	37	$ 1 - 3.65T + 37T^{2} $
	41	$ 1 - 2.58iT - 41T^{2} $
	43	$ 1 - 9.65iT - 43T^{2} $
	47	$ 1 - 12.4T + 47T^{2} $
	53	$ 1 - 5.07iT - 53T^{2} $
	59	$ 1 - 2.34T + 59T^{2} $
	61	$ 1 - 7.65T + 61T^{2} $
	67	$ 1 + 12iT - 67T^{2} $
	71	$ 1 + 4.48T + 71T^{2} $
	73	$ 1 - 4T + 73T^{2} $
	79	$ 1 - 6.48iT - 79T^{2} $
	83	$ 1 + 5.17T + 83T^{2} $
	89	$ 1 - 12.2iT - 89T^{2} $
	97	$ 1 - 13.6T + 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

−9.659782263365283782899747319839, −9.006824808807019217940217507063, −8.265305353930769001933405575848, −7.989776243437302024940850642402, −6.07729898028013508789540023631, −5.77471564526972110438194069593, −4.92732105823723651188326058489, −3.87139355916767467356030409684, −2.46970747226928753047365743098, −1.38847680784358031150558796350, 0.64734792517724004595993496822, 2.53141991830990794282502796096, 3.36945640649224550823612231383, 4.29028695802397280129765324100, 5.42205513459670754041838068018, 6.76024066656569152880103174150, 7.07957111567331679685021720726, 7.67198198379906614342749987224, 8.890880769226326988577320052968, 10.01331724837133053638583730926

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.394543776\)
\(L(\frac12)\)	\(\approx\)	\(1.394543776\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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