LMFDB - L-function 1-240-240.179-r0-0-0

Dirichlet series

L(s) = 1

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

L(s) = 1

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$240$ = $2^{4} \cdot 3 \cdot 5$
Sign:	$0.382 - 0.923i$
Analytic conductor:	$1.11455$
Root analytic conductor:	$1.11455$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{240} (179, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 240,\ (0:\ ),\ 0.382 - 0.923i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8179913775 - 0.5465643644i$
$L(\frac12)$	$\approx$	$0.8179913775 - 0.5465643644i$
$L(1)$	$\approx$	$0.9255553623 - 0.2068141687i$
$L(1)$	$\approx$	$0.9255553623 - 0.2068141687i$

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 - T $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 - iT $
	41	$ 1 $
	43	$ 1 - iT $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 + T $
	67	$ 1 $
	71	$ 1 - iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 + T $
	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

−26.15190749765898726681574360401, −25.57397055034247572071992023905, −24.65985991232625457672635792814, −23.269205593597592849126078812468, −22.94778824093387598606277580794, −21.717512542393328459808081755026, −20.84517494035208035922719620924, −19.77820810877779868925675222714, −18.96454883992551183042288324927, −18.072175547236951136101345203296, −16.72994815093635294046256458765, −16.251685411371243413094899115033, −14.94183823268132981244966016346, −14.14817469067739343819335936206, −12.79506717404215774686839534226, −12.280424435601826505470368736449, −10.914312953723082174110329258038, −9.78023560555355267432743021397, −9.12411298931315809830848283879, −7.59070552961348869572795779873, −6.72917205762816271041628317393, −5.56172289228254004552301182437, −4.21582448339759127435915467088, −3.08835759373921316614357212660, −1.60847144595022416460358667338, 0.739849064957020356426768335863, 2.78557517359073033875421664225, 3.57518561733692363749664924493, 5.231993443398095803011504825059, 6.16549271699470535027648507755, 7.327438240311009893739750044791, 8.50568310426989267554989362471, 9.56160650572696755496255185412, 10.54506955275824459249650229629, 11.602186668077718863957088870425, 12.86096671933584119990289189374, 13.43694810752523968771364938888, 14.73186433173428841512812842450, 15.733970600914965541872646642716, 16.56075638110085335563334464985, 17.51762032108246927399050942233, 18.78156681358589885767477273910, 19.36960133500664622071692062355, 20.420248273828803752057179682655, 21.475318299398398128156326144500, 22.377658215580481820578206361346, 23.16316984729890701729851397392, 24.194305218230777763329711252246, 25.21069262145087570384247973178, 25.9311018812890220116832208078

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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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.8179913775 - 0.5465643644i\)
\(L(\frac12)\)	\(\approx\)	\(0.8179913775 - 0.5465643644i\)
\(L(1)\)	\(\approx\)	\(0.9255553623 - 0.2068141687i\)
\(L(1)\)	\(\approx\)	\(0.9255553623 - 0.2068141687i\)

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