LMFDB - L-function 2-1152-8.3-c2-0-23

Dirichlet series

L(s) = 1

− 5.29i·5^-s + 7.48i·7^-s + 5.65·11^-s − 4i·13^-s − 21.1·17^-s + 29.9·19^-s − 22.6i·23^-s − 3.00·25^-s − 5.29i·29^-s + 22.4i·31^-s + 39.5·35^-s + 28i·37^-s + 63.4·41^-s − 29.9·43^-s − 67.8i·47^-s + ⋯

L(s) = 1

− 1.05i·5^-s + 1.06i·7^-s + 0.514·11^-s − 0.307i·13^-s − 1.24·17^-s + 1.57·19^-s − 0.983i·23^-s − 0.120·25^-s − 0.182i·29^-s + 0.724i·31^-s + 1.13·35^-s + 0.756i·37^-s + 1.54·41^-s − 0.696·43^-s − 1.44i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$31.3897$
Root analytic conductor:	$5.60265$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (703, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1),\ 0.707 + 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.908686082$
$L(\frac12)$	$\approx$	$1.908686082$
$L(2)$		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 + 5.29iT - 25T^{2} $
	7	$ 1 - 7.48iT - 49T^{2} $
	11	$ 1 - 5.65T + 121T^{2} $
	13	$ 1 + 4iT - 169T^{2} $
	17	$ 1 + 21.1T + 289T^{2} $
	19	$ 1 - 29.9T + 361T^{2} $
	23	$ 1 + 22.6iT - 529T^{2} $
	29	$ 1 + 5.29iT - 841T^{2} $
	31	$ 1 - 22.4iT - 961T^{2} $
	37	$ 1 - 28iT - 1.36e3T^{2} $
	41	$ 1 - 63.4T + 1.68e3T^{2} $
	43	$ 1 + 29.9T + 1.84e3T^{2} $
	47	$ 1 + 67.8iT - 2.20e3T^{2} $
	53	$ 1 - 47.6iT - 2.80e3T^{2} $
	59	$ 1 - 101.T + 3.48e3T^{2} $
	61	$ 1 + 76iT - 3.72e3T^{2} $
	67	$ 1 + 59.8T + 4.48e3T^{2} $
	71	$ 1 + 90.5iT - 5.04e3T^{2} $
	73	$ 1 - 26T + 5.32e3T^{2} $
	79	$ 1 + 127. iT - 6.24e3T^{2} $
	83	$ 1 - 118.T + 6.88e3T^{2} $
	89	$ 1 + 42.3T + 7.92e3T^{2} $
	97	$ 1 - 18T + 9.40e3T^{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

−9.191862500100757790870548624849, −8.859049871230294393473653183424, −8.098966292591681645802254191153, −6.96348067499136555681067592805, −6.02241350821158002025955337022, −5.15931633936319713759785409658, −4.49806971085420248470725414937, −3.17348423941846836989068791107, −2.01177067866957031477257909464, −0.71396711984107043857150220849, 1.02864294030720952459993727424, 2.46827739988855517484640408431, 3.57445486295418085256722010699, 4.28278064950008385723380400906, 5.54502309055337924699080904041, 6.61487777480005039099859953196, 7.16082590635146067101136517507, 7.78789026351528471565252079573, 9.089615272813618628429298752896, 9.724122004550450320114891404802

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.908686082\)
\(L(\frac12)\)	\(\approx\)	\(1.908686082\)
\(L(2)\)		not available
\(L(1)\)		not available

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