LMFDB - L-function 2-117-13.12-c3-0-13

Dirichlet series

L(s) = 1

− 1.32i·2^-s + 6.23·4^-s − 15.4i·5^-s + 7.96i·7^-s − 18.9i·8^-s − 20.4·10^-s + 12.7i·11^-s + (7.47 − 46.2i)13^-s + 10.5·14^-s + 24.8·16^-s − 54·17^-s − 84.5i·19^-s − 96.2i·20^-s + 16.9·22^-s + 122.·23^-s + ⋯

L(s) = 1

− 0.469i·2^-s + 0.779·4^-s − 1.37i·5^-s + 0.430i·7^-s − 0.835i·8^-s − 0.647·10^-s + 0.350i·11^-s + (0.159 − 0.987i)13^-s + 0.201·14^-s + 0.387·16^-s − 0.770·17^-s − 1.02i·19^-s − 1.07i·20^-s + 0.164·22^-s + 1.11·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$117$ = $3^{2} \cdot 13$
Sign:	$-0.159 + 0.987i$
Analytic conductor:	$6.90322$
Root analytic conductor:	$2.62739$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{117} (64, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 117,\ (\ :3/2),\ -0.159 + 0.987i)$

Particular Values

$L(2)$	$\approx$	$1.20047 - 1.40998i$
$L(\frac12)$	$\approx$	$1.20047 - 1.40998i$
$L(\frac{5}{2})$		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 $
	13	$ 1 + (-7.47 + 46.2i)T $
good	2	$ 1 + 1.32iT - 8T^{2} $
	5	$ 1 + 15.4iT - 125T^{2} $
	7	$ 1 - 7.96iT - 343T^{2} $
	11	$ 1 - 12.7iT - 1.33e3T^{2} $
	17	$ 1 + 54T + 4.91e3T^{2} $
	19	$ 1 + 84.5iT - 6.85e3T^{2} $
	23	$ 1 - 122.T + 1.21e4T^{2} $
	29	$ 1 + 140.T + 2.43e4T^{2} $
	31	$ 1 + 116. iT - 2.97e4T^{2} $
	37	$ 1 - 433. iT - 5.06e4T^{2} $
	41	$ 1 - 205. iT - 6.89e4T^{2} $
	43	$ 1 - 418.T + 7.95e4T^{2} $
	47	$ 1 - 485. iT - 1.03e5T^{2} $
	53	$ 1 - 674.T + 1.48e5T^{2} $
	59	$ 1 - 186. iT - 2.05e5T^{2} $
	61	$ 1 + 671.T + 2.26e5T^{2} $
	67	$ 1 - 14.0iT - 3.00e5T^{2} $
	71	$ 1 - 346. iT - 3.57e5T^{2} $
	73	$ 1 - 832. iT - 3.89e5T^{2} $
	79	$ 1 + 335.T + 4.93e5T^{2} $
	83	$ 1 + 568. iT - 5.71e5T^{2} $
	89	$ 1 + 236. iT - 7.04e5T^{2} $
	97	$ 1 + 1.27e3iT - 9.12e5T^{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

−12.80471218380316483236689017640, −11.76977748032808340142275505743, −10.86609103508001420155657237941, −9.554563662502972537838976732652, −8.615967190490185903399101696328, −7.30500279823175762453336620302, −5.83292012781761417997701667359, −4.55803331966515885335514194982, −2.70537249330480534237819096703, −1.04219626030381770724059870583, 2.21358440604302909228279161300, 3.70510994033716105615211728630, 5.78661490425240969965153329397, 6.87213067379417310450204450260, 7.40912475143256646982549569631, 8.970739713369947696623887366944, 10.65005991566131917565059133257, 10.95480501464025518972256874935, 12.10794185675480091614875837619, 13.70101119832768932846734489477

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.20047 - 1.40998i\)
\(L(\frac12)\)	\(\approx\)	\(1.20047 - 1.40998i\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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