LMFDB - L-function 2-960-20.19-c2-0-44

Dirichlet series

L(s) = 1

+ 1.73·3^-s + (3.27 − 3.77i)5^-s − 9.55·7^-s + 2.99·9^-s − 9.92i·11^-s + 7.55i·13^-s + (5.67 − 6.54i)15^-s + 17.1i·17^-s − 26.1i·19^-s − 16.5·21^-s + 1.67·23^-s + (−3.54 − 24.7i)25^-s + 5.19·27^-s + 0.350·29^-s − 46.0i·31^-s + ⋯

L(s) = 1

+ 0.577·3^-s + (0.654 − 0.755i)5^-s − 1.36·7^-s + 0.333·9^-s − 0.902i·11^-s + 0.581i·13^-s + (0.378 − 0.436i)15^-s + 1.01i·17^-s − 1.37i·19^-s − 0.788·21^-s + 0.0728·23^-s + (−0.141 − 0.989i)25^-s + 0.192·27^-s + 0.0120·29^-s − 1.48i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$960$ = $2^{6} \cdot 3 \cdot 5$
Sign:	$-0.654 + 0.755i$
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:	$(2,\ 960,\ (\ :1),\ -0.654 + 0.755i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.415680291$
$L(\frac12)$	$\approx$	$1.415680291$
$L(2)$		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 + (-3.27 + 3.77i)T $
good	7	$ 1 + 9.55T + 49T^{2} $
	11	$ 1 + 9.92iT - 121T^{2} $
	13	$ 1 - 7.55iT - 169T^{2} $
	17	$ 1 - 17.1iT - 289T^{2} $
	19	$ 1 + 26.1iT - 361T^{2} $
	23	$ 1 - 1.67T + 529T^{2} $
	29	$ 1 - 0.350T + 841T^{2} $
	31	$ 1 + 46.0iT - 961T^{2} $
	37	$ 1 + 22.6iT - 1.36e3T^{2} $
	41	$ 1 + 77.2T + 1.68e3T^{2} $
	43	$ 1 + 41.7T + 1.84e3T^{2} $
	47	$ 1 + 14.0T + 2.20e3T^{2} $
	53	$ 1 + 22.6iT - 2.80e3T^{2} $
	59	$ 1 + 94.7iT - 3.48e3T^{2} $
	61	$ 1 + 38T + 3.72e3T^{2} $
	67	$ 1 + 29.8T + 4.48e3T^{2} $
	71	$ 1 - 7.19iT - 5.04e3T^{2} $
	73	$ 1 - 34.3iT - 5.32e3T^{2} $
	79	$ 1 - 46.0iT - 6.24e3T^{2} $
	83	$ 1 - 24.1T + 6.88e3T^{2} $
	89	$ 1 - 100.T + 7.92e3T^{2} $
	97	$ 1 - 131. iT - 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

−9.448620514679004092419652091457, −8.833943184329076697101537714829, −8.108294310645175620239935838485, −6.75435692656009307073068749408, −6.23878174848819450629284981161, −5.18066519581018952283851599399, −4.00085748818378785444668807247, −3.07373298256087246620365470848, −1.91940676045167751169790719887, −0.38814877374767682492841676367, 1.69725631328386866928097046140, 2.96198889266008129213025067785, 3.42380788583507495960345614882, 4.91140165183307853716732443304, 6.01338291814520598123951121236, 6.82707877948849577259185419185, 7.41073315166320810766993754284, 8.580454647714336838157534800465, 9.534938660079355877592139527049, 10.07040393847998904103459892242

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.415680291\)
\(L(\frac12)\)	\(\approx\)	\(1.415680291\)
\(L(2)\)		not available
\(L(1)\)		not available

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