LMFDB - L-function 2-1143-3.2-c2-0-27

Dirichlet series

L(s) = 1

− 2.99i·2^-s − 4.95·4^-s + 7.40i·5^-s − 4.26·7^-s + 2.85i·8^-s + 22.1·10^-s − 11.1i·11^-s − 7.91·13^-s + 12.7i·14^-s − 11.2·16^-s − 14.3i·17^-s + 18.0·19^-s − 36.6i·20^-s − 33.4·22^-s + 33.5i·23^-s + ⋯

L(s) = 1

− 1.49i·2^-s − 1.23·4^-s + 1.48i·5^-s − 0.608·7^-s + 0.356i·8^-s + 2.21·10^-s − 1.01i·11^-s − 0.609·13^-s + 0.910i·14^-s − 0.704·16^-s − 0.842i·17^-s + 0.952·19^-s − 1.83i·20^-s − 1.52·22^-s + 1.45i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1143$ = $3^{2} \cdot 127$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$31.1444$
Root analytic conductor:	$5.58072$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1143} (890, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1143,\ (\ :1),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.499278544$
$L(\frac12)$	$\approx$	$1.499278544$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	127	$ 1 - 11.2T $
good	2	$ 1 + 2.99iT - 4T^{2} $
	5	$ 1 - 7.40iT - 25T^{2} $
	7	$ 1 + 4.26T + 49T^{2} $
	11	$ 1 + 11.1iT - 121T^{2} $
	13	$ 1 + 7.91T + 169T^{2} $
	17	$ 1 + 14.3iT - 289T^{2} $
	19	$ 1 - 18.0T + 361T^{2} $
	23	$ 1 - 33.5iT - 529T^{2} $
	29	$ 1 - 23.5iT - 841T^{2} $
	31	$ 1 - 43.9T + 961T^{2} $
	37	$ 1 - 2.75T + 1.36e3T^{2} $
	41	$ 1 - 16.3iT - 1.68e3T^{2} $
	43	$ 1 - 48.3T + 1.84e3T^{2} $
	47	$ 1 - 4.06iT - 2.20e3T^{2} $
	53	$ 1 - 74.2iT - 2.80e3T^{2} $
	59	$ 1 + 55.4iT - 3.48e3T^{2} $
	61	$ 1 - 45.3T + 3.72e3T^{2} $
	67	$ 1 - 124.T + 4.48e3T^{2} $
	71	$ 1 + 116. iT - 5.04e3T^{2} $
	73	$ 1 - 93.3T + 5.32e3T^{2} $
	79	$ 1 + 2.92T + 6.24e3T^{2} $
	83	$ 1 + 72.3iT - 6.88e3T^{2} $
	89	$ 1 - 19.0iT - 7.92e3T^{2} $
	97	$ 1 - 18.2T + 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

−9.698865640022206838866842017376, −9.239184348437023643249747922595, −7.81160025860486015970644451596, −6.99245657384255261472932787444, −6.18267396928716147158970674117, −4.98716513809282528806806901305, −3.52384724438803910444224202281, −3.16174279775985762844948579495, −2.39210465440827771377052498115, −0.844640069844061941870258468343, 0.63234404710714892878461884103, 2.30344090311794408347376808586, 4.15546437648124963285171801725, 4.79150020252709150412021817220, 5.54151755564813062741736280478, 6.44737800009277035834935850696, 7.19845915848023733802037391629, 8.158096388543582348103743492541, 8.568846845522775706489225657487, 9.592006980527911119550900430843

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.499278544\)
\(L(\frac12)\)	\(\approx\)	\(1.499278544\)
\(L(2)\)		not available
\(L(1)\)		not available

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