LMFDB - L-function 1-240-240.203-r1-0-0

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$240$ = $2^{4} \cdot 3 \cdot 5$
Sign:	$0.584 - 0.811i$
Analytic conductor:	$25.7915$
Root analytic conductor:	$25.7915$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{240} (203, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 240,\ (1:\ ),\ 0.584 - 0.811i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.635299016 - 0.8371394824i$
$L(\frac12)$	$\approx$	$1.635299016 - 0.8371394824i$
$L(1)$	$\approx$	$1.132334851 - 0.1837526503i$
$L(1)$	$\approx$	$1.132334851 - 0.1837526503i$

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

−25.91551865155845505465711863110, −25.220473921467708583247442241134, −24.33582004063397806584274799811, −23.348265256082797725584061520253, −22.3456293114110479096347616038, −21.459615721221380374144874072957, −20.70832683954966057684908459433, −19.46544240469290585812154536853, −18.53415738638468113379032928286, −17.97467583438726641739159812349, −16.40320286901007101012210248880, −15.94265373662906456118329303325, −14.7196097634218867483583651545, −13.776835034792682778489168165731, −12.73215287712551204132785147624, −11.63879636835045480775313580456, −10.87606663623303646444782334960, −9.4127945155040995748423764259, −8.67287561409507563396018740962, −7.55083718207954576487158874209, −6.06330952045021560882620802710, −5.44054623464780956599534578108, −3.80516048151390820815877562248, −2.7062878592133200723299586384, −1.17435624206612555503805095663, 0.691730328213957375716542649984, 2.13064428244546652993171754803, 3.76434797227200077707919640544, 4.58838657056320703633670128927, 6.119356949456832656512220751710, 7.112628980026707230692422115450, 8.14573538973047058714189378339, 9.36746434828114778488372064381, 10.4700350196212049481224214758, 11.21029847652039630258621583399, 12.63335553227036713024693856642, 13.35455142925447921463580352559, 14.46336163613413856142716413012, 15.40553763219103385067247694614, 16.50330692783588430690027345388, 17.39639005070992714358944701634, 18.222830080685111104321817468, 19.47045981392231103797943017698, 20.26578768046850379216804104585, 21.04186471354191261241415142898, 22.231079335132232689967262998881, 23.206392069070179786789647584295, 23.771530394091276565977493151022, 24.94666945802999037445929107953, 26.04965978006639919169387527608

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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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.635299016 - 0.8371394824i\)
\(L(\frac12)\)	\(\approx\)	\(1.635299016 - 0.8371394824i\)
\(L(1)\)	\(\approx\)	\(1.132334851 - 0.1837526503i\)
\(L(1)\)	\(\approx\)	\(1.132334851 - 0.1837526503i\)

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