LMFDB - L-function 2-1143-3.2-c2-0-28

Dirichlet series

L(s) = 1

− 3.63i·2^-s − 9.24·4^-s − 3.49i·5^-s − 9.62·7^-s + 19.0i·8^-s − 12.7·10^-s + 19.0i·11^-s + 20.1·13^-s + 35.0i·14^-s + 32.4·16^-s − 0.819i·17^-s + 7.28·19^-s + 32.2i·20^-s + 69.3·22^-s + 0.446i·23^-s + ⋯

L(s) = 1

− 1.81i·2^-s − 2.31·4^-s − 0.698i·5^-s − 1.37·7^-s + 2.38i·8^-s − 1.27·10^-s + 1.73i·11^-s + 1.55·13^-s + 2.50i·14^-s + 2.02·16^-s − 0.0482i·17^-s + 0.383·19^-s + 1.61i·20^-s + 3.15·22^-s + 0.0194i·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

$L(\frac{3}{2})$	$\approx$	$1.271264117$
$L(\frac12)$	$\approx$	$1.271264117$
$L(2)$		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 + 3.63iT - 4T^{2} $
	5	$ 1 + 3.49iT - 25T^{2} $
	7	$ 1 + 9.62T + 49T^{2} $
	11	$ 1 - 19.0iT - 121T^{2} $
	13	$ 1 - 20.1T + 169T^{2} $
	17	$ 1 + 0.819iT - 289T^{2} $
	19	$ 1 - 7.28T + 361T^{2} $
	23	$ 1 - 0.446iT - 529T^{2} $
	29	$ 1 + 43.3iT - 841T^{2} $
	31	$ 1 + 12.5T + 961T^{2} $
	37	$ 1 + 31.1T + 1.36e3T^{2} $
	41	$ 1 + 17.5iT - 1.68e3T^{2} $
	43	$ 1 - 21.4T + 1.84e3T^{2} $
	47	$ 1 - 61.7iT - 2.20e3T^{2} $
	53	$ 1 - 65.6iT - 2.80e3T^{2} $
	59	$ 1 - 15.3iT - 3.48e3T^{2} $
	61	$ 1 - 59.7T + 3.72e3T^{2} $
	67	$ 1 + 51.8T + 4.48e3T^{2} $
	71	$ 1 - 11.6iT - 5.04e3T^{2} $
	73	$ 1 - 123.T + 5.32e3T^{2} $
	79	$ 1 - 52.3T + 6.24e3T^{2} $
	83	$ 1 + 92.4iT - 6.88e3T^{2} $
	89	$ 1 - 1.56iT - 7.92e3T^{2} $
	97	$ 1 + 40.8T + 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

−9.305599529583338468433224770423, −9.226741281755248545042374105631, −8.036129433318338351008939609740, −6.78132924481078968430276713770, −5.65740127163889840124714863109, −4.51408789919448288491372964415, −3.87598656122222219996762391127, −2.91957065286895074728295953178, −1.81234645591024138623824341219, −0.72990888225599449133846573678, 0.64719835705068402446625967618, 3.34369871048966778375225699865, 3.60329442687139637569197255405, 5.28519770119951189399109669671, 5.96658352865322080822215772417, 6.59586145315964119648028465212, 7.05260255392077277460708374191, 8.330012115025341299747152845305, 8.708322752696939391622836825054, 9.525765511087029749350036749802

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.271264117\)
\(L(\frac12)\)	\(\approx\)	\(1.271264117\)
\(L(2)\)		not available
\(L(1)\)		not available

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