LMFDB - L-function 2-960-20.19-c2-0-29

Dirichlet series

L(s) = 1

− 1.73·3^-s + (4.13 − 2.81i)5^-s − 5.81·7^-s + 2.99·9^-s + 6.08i·11^-s + 1.49i·13^-s + (−7.15 + 4.87i)15^-s − 18.3i·17^-s + 23.5i·19^-s + 10.0·21^-s + 24.0·23^-s + (9.15 − 23.2i)25^-s − 5.19·27^-s + 44.1·29^-s − 39.3i·31^-s + ⋯

L(s) = 1

− 0.577·3^-s + (0.826 − 0.562i)5^-s − 0.830·7^-s + 0.333·9^-s + 0.552i·11^-s + 0.114i·13^-s + (−0.477 + 0.325i)15^-s − 1.07i·17^-s + 1.24i·19^-s + 0.479·21^-s + 1.04·23^-s + (0.366 − 0.930i)25^-s − 0.192·27^-s + 1.52·29^-s − 1.26i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$960$ = $2^{6} \cdot 3 \cdot 5$
Sign:	$0.562 + 0.826i$
Analytic conductor:	$26.1581$
Root analytic conductor:	$5.11449$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{960} (319, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 960,\ (\ :1),\ 0.562 + 0.826i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.510015651$
$L(\frac12)$	$\approx$	$1.510015651$
$L(2)$		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 + 1.73T $
	5	$ 1 + (-4.13 + 2.81i)T $
good	7	$ 1 + 5.81T + 49T^{2} $
	11	$ 1 - 6.08iT - 121T^{2} $
	13	$ 1 - 1.49iT - 169T^{2} $
	17	$ 1 + 18.3iT - 289T^{2} $
	19	$ 1 - 23.5iT - 361T^{2} $
	23	$ 1 - 24.0T + 529T^{2} $
	29	$ 1 - 44.1T + 841T^{2} $
	31	$ 1 + 39.3iT - 961T^{2} $
	37	$ 1 - 24.8iT - 1.36e3T^{2} $
	41	$ 1 + 43.1T + 1.68e3T^{2} $
	43	$ 1 + 40.3T + 1.84e3T^{2} $
	47	$ 1 - 35.6T + 2.20e3T^{2} $
	53	$ 1 + 75.2iT - 2.80e3T^{2} $
	59	$ 1 + 56.3iT - 3.48e3T^{2} $
	61	$ 1 - 77.0T + 3.72e3T^{2} $
	67	$ 1 - 69.2T + 4.48e3T^{2} $
	71	$ 1 + 37.4iT - 5.04e3T^{2} $
	73	$ 1 + 53.6iT - 5.32e3T^{2} $
	79	$ 1 + 82.9iT - 6.24e3T^{2} $
	83	$ 1 + 8.44T + 6.88e3T^{2} $
	89	$ 1 - 94.1T + 7.92e3T^{2} $
	97	$ 1 + 89.4iT - 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

−9.976186496750807896870519734953, −9.066758902113087821713044016639, −8.070251135773129011617971984298, −6.85861996151991235847679232507, −6.33502746146118695433027822767, −5.30597567023629285635172411378, −4.64700819260630929801226895401, −3.27788018645608669266130867239, −1.97675497884828739376374353535, −0.62882076894476466723851726145, 1.04208631654662544415897567718, 2.59237591598111883591086429322, 3.48115894777272890080588076151, 4.89076320600108238408677327417, 5.74896669858095518136016877132, 6.62524986700134464486774598328, 6.97346775683595808014146630146, 8.471055375205968587269459785144, 9.197244848187931887651634809909, 10.18957473159206828635959981966

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(1.510015651\)
\(L(\frac12)\)	\(\approx\)	\(1.510015651\)
\(L(2)\)		not available
\(L(1)\)		not available

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