LMFDB - L-function 1-240-240.77-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

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

Invariants

Degree:	$1$
Conductor:	$240$ = $2^{4} \cdot 3 \cdot 5$
Sign:	$0.811 - 0.584i$
Analytic conductor:	$1.11455$
Root analytic conductor:	$1.11455$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{240} (77, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 240,\ (0:\ ),\ 0.811 - 0.584i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.165301054 - 0.3761857562i$
$L(\frac12)$	$\approx$	$1.165301054 - 0.3761857562i$
$L(1)$	$\approx$	$1.086321679 - 0.1565124024i$
$L(1)$	$\approx$	$1.086321679 - 0.1565124024i$

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

−26.220382064257015913646353528744, −25.313637484579677270347951111276, −24.526581113623232116180490936004, −23.54215828469549828087022308010, −22.58986240660801866234101722540, −21.44438911889570355351838575040, −21.079858867356359854388926732296, −19.57085941084321155950921032861, −18.84817041287311207474443933991, −18.01884641814183110831082291452, −16.83498273831770704114383783394, −15.865803803193787555818179400022, −15.066346111013513985098982145142, −13.90182808491216382292626928898, −12.97062855840641409999074205099, −11.84718801549491243705507678741, −11.02484286307279456674151060180, −9.78627046137616971993549843509, −8.62947243043469988804899110435, −7.96451688598810200717962301684, −6.15716002377605250846110977318, −5.7506082120748285042007523227, −4.07633315396502243284125313163, −2.9459470046296532838002711612, −1.47754921937299700719787944741, 1.03022770119630283725345199408, 2.65563838318113078372092398857, 4.06293875711406236777354909626, 4.96850547342363034422630019954, 6.56807993653776378044060312572, 7.29999129135756840778744168824, 8.55242984379699403586801252901, 9.71700436738572592892992104016, 10.665051338305563479219745251070, 11.62602451410469203866110896726, 12.899631153879954212038234924469, 13.68482146335122702141021886387, 14.69730784605379660708218876964, 15.820708112109810431818334890982, 16.68386166430663181383142419552, 17.75082144197663660264814580141, 18.46874773542171334604233715100, 19.902179355592794346962388199556, 20.36025248887901328061742285571, 21.36671649485440195641028379322, 22.701055806746891399357852491830, 23.18393632008777854287205736141, 24.1925833329323869664526877964, 25.284798755151197244201041203662, 26.116990129171269696674054284484

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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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.165301054 - 0.3761857562i\)
\(L(\frac12)\)	\(\approx\)	\(1.165301054 - 0.3761857562i\)
\(L(1)\)	\(\approx\)	\(1.086321679 - 0.1565124024i\)
\(L(1)\)	\(\approx\)	\(1.086321679 - 0.1565124024i\)

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