LMFDB - L-function 2-1143-3.2-c2-0-3

Dirichlet series

L(s) = 1

− 3.57i·2^-s − 8.78·4^-s + 5.13i·5^-s + 4.24·7^-s + 17.1i·8^-s + 18.3·10^-s + 2.99i·11^-s − 1.37·13^-s − 15.1i·14^-s + 26.0·16^-s − 29.5i·17^-s − 26.2·19^-s − 45.1i·20^-s + 10.7·22^-s + 1.34i·23^-s + ⋯

L(s) = 1

− 1.78i·2^-s − 2.19·4^-s + 1.02i·5^-s + 0.606·7^-s + 2.14i·8^-s + 1.83·10^-s + 0.272i·11^-s − 0.105·13^-s − 1.08i·14^-s + 1.63·16^-s − 1.74i·17^-s − 1.38·19^-s − 2.25i·20^-s + 0.486·22^-s + 0.0585i·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

$L(\frac{3}{2})$	$\approx$	$0.3019555881$
$L(\frac12)$	$\approx$	$0.3019555881$
$L(2)$		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 + 3.57iT - 4T^{2} $
	5	$ 1 - 5.13iT - 25T^{2} $
	7	$ 1 - 4.24T + 49T^{2} $
	11	$ 1 - 2.99iT - 121T^{2} $
	13	$ 1 + 1.37T + 169T^{2} $
	17	$ 1 + 29.5iT - 289T^{2} $
	19	$ 1 + 26.2T + 361T^{2} $
	23	$ 1 - 1.34iT - 529T^{2} $
	29	$ 1 + 4.79iT - 841T^{2} $
	31	$ 1 - 40.7T + 961T^{2} $
	37	$ 1 + 54.4T + 1.36e3T^{2} $
	41	$ 1 - 59.8iT - 1.68e3T^{2} $
	43	$ 1 + 36.0T + 1.84e3T^{2} $
	47	$ 1 - 13.5iT - 2.20e3T^{2} $
	53	$ 1 - 57.2iT - 2.80e3T^{2} $
	59	$ 1 - 94.2iT - 3.48e3T^{2} $
	61	$ 1 + 95.2T + 3.72e3T^{2} $
	67	$ 1 + 4.97T + 4.48e3T^{2} $
	71	$ 1 + 54.4iT - 5.04e3T^{2} $
	73	$ 1 + 52.2T + 5.32e3T^{2} $
	79	$ 1 + 31.2T + 6.24e3T^{2} $
	83	$ 1 + 5.29iT - 6.88e3T^{2} $
	89	$ 1 + 6.08iT - 7.92e3T^{2} $
	97	$ 1 + 129.T + 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

−10.09510375239748728517027331830, −9.236011533082748580199954809967, −8.417868754001491592448337545264, −7.36822470323917917933640812689, −6.39253623165365050719546290577, −4.93314950713525633137043308285, −4.37026624325889895939816528115, −3.08049246701096213700966493656, −2.56986887609633633532358238176, −1.42695595734330366173671175238, 0.092869666675949189279105727059, 1.66802814574898532335168322455, 3.84307135004625043023065647017, 4.65464691272929640493922855175, 5.30996636410821341796349251722, 6.19806928573946740817535095118, 6.87457867037241667763866812283, 8.093084453177789533963874701095, 8.421243433733171632190958431035, 8.879336709905565315431417133451

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.3019555881\)
\(L(\frac12)\)	\(\approx\)	\(0.3019555881\)
\(L(2)\)		not available
\(L(1)\)		not available

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