LMFDB - L-function 2-1143-3.2-c2-0-70

Dirichlet series

L(s) = 1

− 2.40i·2^-s − 1.80·4^-s + 4.01i·5^-s − 8.05·7^-s − 5.28i·8^-s + 9.67·10^-s − 7.16i·11^-s + 20.5·13^-s + 19.4i·14^-s − 19.9·16^-s + 13.1i·17^-s − 14.0·19^-s − 7.25i·20^-s − 17.2·22^-s − 26.8i·23^-s + ⋯

L(s) = 1

− 1.20i·2^-s − 0.451·4^-s + 0.802i·5^-s − 1.15·7^-s − 0.660i·8^-s + 0.967·10^-s − 0.651i·11^-s + 1.57·13^-s + 1.38i·14^-s − 1.24·16^-s + 0.771i·17^-s − 0.740·19^-s − 0.362i·20^-s − 0.785·22^-s − 1.16i·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

$L(\frac{3}{2})$	$\approx$	$0.4228005439$
$L(\frac12)$	$\approx$	$0.4228005439$
$L(2)$		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 + 2.40iT - 4T^{2} $
	5	$ 1 - 4.01iT - 25T^{2} $
	7	$ 1 + 8.05T + 49T^{2} $
	11	$ 1 + 7.16iT - 121T^{2} $
	13	$ 1 - 20.5T + 169T^{2} $
	17	$ 1 - 13.1iT - 289T^{2} $
	19	$ 1 + 14.0T + 361T^{2} $
	23	$ 1 + 26.8iT - 529T^{2} $
	29	$ 1 - 1.48iT - 841T^{2} $
	31	$ 1 + 47.6T + 961T^{2} $
	37	$ 1 + 50.2T + 1.36e3T^{2} $
	41	$ 1 + 54.4iT - 1.68e3T^{2} $
	43	$ 1 - 38.8T + 1.84e3T^{2} $
	47	$ 1 - 59.0iT - 2.20e3T^{2} $
	53	$ 1 - 50.8iT - 2.80e3T^{2} $
	59	$ 1 + 36.7iT - 3.48e3T^{2} $
	61	$ 1 + 111.T + 3.72e3T^{2} $
	67	$ 1 + 37.8T + 4.48e3T^{2} $
	71	$ 1 + 60.7iT - 5.04e3T^{2} $
	73	$ 1 + 96.3T + 5.32e3T^{2} $
	79	$ 1 + 148.T + 6.24e3T^{2} $
	83	$ 1 + 100. iT - 6.88e3T^{2} $
	89	$ 1 - 123. iT - 7.92e3T^{2} $
	97	$ 1 + 69.7T + 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

−9.129748119101394230822317261880, −8.699712623229143792388094953699, −7.27943774310570654426470323721, −6.38068645812892320107998473437, −5.99695213717301214090945817958, −4.14257078895677532932553500382, −3.44538205385218232510482160347, −2.81511981606883908497293131163, −1.56355270074461139276141541000, −0.12188657412906185730500435546, 1.61186624369181366134812184291, 3.18008770869214042644030274579, 4.29268847422937681232742433441, 5.35681145771244031877096583123, 6.00227705403508929989945009671, 6.85107728658410378078284882379, 7.45870588861996713197172208425, 8.615376539715544318572713381371, 8.958112210199822734561292713661, 9.859657701723967859407178306633

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.4228005439\)
\(L(\frac12)\)	\(\approx\)	\(0.4228005439\)
\(L(2)\)		not available
\(L(1)\)		not available

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