LMFDB - L-function 2-1152-12.11-c1-0-1

Dirichlet series

L(s) = 1

− 0.585i·5^-s + 0.828i·7^-s − 2.82·11^-s − 2.82·13^-s + 2.58i·17^-s + 5.65i·19^-s − 6.82·23^-s + 4.65·25^-s − 3.41i·29^-s + 8.82i·31^-s + 0.485·35^-s − 7.65·37^-s + 5.41i·41^-s + 1.65i·43^-s + 4.48·47^-s + ⋯

L(s) = 1

− 0.261i·5^-s + 0.313i·7^-s − 0.852·11^-s − 0.784·13^-s + 0.627i·17^-s + 1.29i·19^-s − 1.42·23^-s + 0.931·25^-s − 0.634i·29^-s + 1.58i·31^-s + 0.0820·35^-s − 1.25·37^-s + 0.845i·41^-s + 0.252i·43^-s + 0.654·47^-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

$L(1)$	$\approx$	$0.7159483209$
$L(\frac12)$	$\approx$	$0.7159483209$
$L(\frac{3}{2})$		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 + 0.585iT - 5T^{2} $
	7	$ 1 - 0.828iT - 7T^{2} $
	11	$ 1 + 2.82T + 11T^{2} $
	13	$ 1 + 2.82T + 13T^{2} $
	17	$ 1 - 2.58iT - 17T^{2} $
	19	$ 1 - 5.65iT - 19T^{2} $
	23	$ 1 + 6.82T + 23T^{2} $
	29	$ 1 + 3.41iT - 29T^{2} $
	31	$ 1 - 8.82iT - 31T^{2} $
	37	$ 1 + 7.65T + 37T^{2} $
	41	$ 1 - 5.41iT - 41T^{2} $
	43	$ 1 - 1.65iT - 43T^{2} $
	47	$ 1 - 4.48T + 47T^{2} $
	53	$ 1 + 9.07iT - 53T^{2} $
	59	$ 1 + 13.6T + 59T^{2} $
	61	$ 1 + 3.65T + 61T^{2} $
	67	$ 1 - 12iT - 67T^{2} $
	71	$ 1 + 12.4T + 71T^{2} $
	73	$ 1 - 4T + 73T^{2} $
	79	$ 1 - 10.4iT - 79T^{2} $
	83	$ 1 - 10.8T + 83T^{2} $
	89	$ 1 - 3.75iT - 89T^{2} $
	97	$ 1 - 2.34T + 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

−10.29317110179427784887164740436, −9.257192126274612856138578594839, −8.320499599362063731320412375970, −7.81740640802887611826150140983, −6.73760494218947621075126231340, −5.76347387890423002773379728221, −5.04339367096477744846169214714, −3.99081539792909957437747345814, −2.82430964362749475080910665349, −1.67208705540559493262913110002, 0.29286805446619933560333623045, 2.19638920872998885645758637939, 3.08367683441855913665075854293, 4.37996645894102702388425627831, 5.15242546716902508036397897380, 6.15686073907591485825550347302, 7.26824704243720903316685394048, 7.59459710264408908853655607753, 8.796675640253452903907022083541, 9.499268722837190812756418712633

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7159483209\)
\(L(\frac12)\)	\(\approx\)	\(0.7159483209\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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