LMFDB - L-function 2-1152-12.11-c1-0-14

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

$L(1)$	$\approx$	$1.616177962$
$L(\frac12)$	$\approx$	$1.616177962$
$L(\frac{3}{2})$		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

−9.437706038300084413516386713056, −8.652408617807454950946310375541, −8.039592069650999901534996369624, −6.99974915433235977548441406420, −6.23560834969160719434475012130, −5.00029805967431548031282124158, −4.22017826406989044631952144554, −3.65527485532749418032713803890, −1.54190869856130659540332771300, −0.75882814927874498073317092590, 1.91805364102669268549813230830, 2.88210380728742240201039710144, 3.64822522226874651074227129262, 5.12113368635456465437296374761, 6.23282334091230663456225298784, 6.38429287030864020589896875492, 7.63976711841726707390573560959, 8.443983538437533135179292021629, 9.416024848594289385158154012154, 9.893816773756206006152652069949

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.616177962\)
\(L(\frac12)\)	\(\approx\)	\(1.616177962\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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