LMFDB - L-function 1-240-240.197-r0-0-0

Dirichlet series

L(s) = 1

− i·7^-s − i·11^-s − 13^-s − i·17^-s + i·19^-s − i·23^-s − i·29^-s + 31^-s − 37^-s + 41^-s + 43^-s − i·47^-s − 49^-s − 53^-s + i·59^-s + ⋯

L(s) = 1

− i·7^-s − i·11^-s − 13^-s − i·17^-s + i·19^-s − i·23^-s − i·29^-s + 31^-s − 37^-s + 41^-s + 43^-s − i·47^-s − 49^-s − 53^-s + i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$240$ = $2^{4} \cdot 3 \cdot 5$
Sign:	$0.160 - 0.987i$
Analytic conductor:	$1.11455$
Root analytic conductor:	$1.11455$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{240} (197, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 240,\ (0:\ ),\ 0.160 - 0.987i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7779974586 - 0.6619231919i$
$L(\frac12)$	$\approx$	$0.7779974586 - 0.6619231919i$
$L(1)$	$\approx$	$0.9398727970 - 0.2765568270i$
$L(1)$	$\approx$	$0.9398727970 - 0.2765568270i$

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 $
	11	$ 1 $
	13	$ 1 $
	17	$ 1 $
	19	$ 1 - iT $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 - iT $
	41	$ 1 $
	43	$ 1 - T $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 - iT $
	67	$ 1 $
	71	$ 1 + iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 - iT $
	97	$ 1 $
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$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−26.20113989833723019321734090302, −25.5445083816088713881409836052, −24.54819004099251108168278654191, −23.76103868239186191640049160652, −22.5406966434422838014432475930, −21.850929280685467527415474983832, −20.963803539947893871511576931667, −19.71128802180286809459664477119, −19.127631160410463356786763281512, −17.765044856108526142351836795868, −17.3316096311499113836906407482, −15.80035045947763274633715059554, −15.164169619363698516264806967576, −14.23881989891650826041391055868, −12.7865583102077099250989329352, −12.24087153091555569991707007800, −11.07580547390285887188592324820, −9.82196965709380468024791455763, −9.02435303858899008156216271116, −7.79386118752900662826514429494, −6.71986897470070597402540468338, −5.461902432762726101869308030813, −4.49899425401802655823752884321, −2.90978271348126433483145217539, −1.81621168787584073550307793274, 0.73734661626703111282374998275, 2.531852972823107683238800694989, 3.816459404169639044327980366796, 4.937601191907071202438777508735, 6.25268120978319289945514510708, 7.35816250438037665328617016271, 8.29284512578200698547194900550, 9.64428539337236868502041594556, 10.498666867120920843407140396771, 11.55830070359049145791310703468, 12.61298643298470873549324807024, 13.85573289863894248049945769372, 14.32484926466040086258934702514, 15.76304829094414879510587218518, 16.66815231526013525031053688038, 17.374709536267404744437269794749, 18.6532799785335198095108292325, 19.44082967317184903779400967968, 20.48479559913209620540411750612, 21.22710003133663683596402629573, 22.48762587832674042193650916485, 23.08102906689984220369788622684, 24.35836737366173054276682830218, 24.775378108028557365592240256252, 26.27694313716831214067504686946

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7779974586 - 0.6619231919i\)
\(L(\frac12)\)	\(\approx\)	\(0.7779974586 - 0.6619231919i\)
\(L(1)\)	\(\approx\)	\(0.9398727970 - 0.2765568270i\)
\(L(1)\)	\(\approx\)	\(0.9398727970 - 0.2765568270i\)

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