LMFDB - L-function 2-4608-12.11-c1-0-45

Dirichlet series

L(s) = 1

− 4.24i·7^-s + 4·11^-s + 6·13^-s − 4.24i·17^-s − 2.82i·19^-s + 6·23^-s + 5·25^-s + 8.48i·29^-s + 4.24i·31^-s + 6·37^-s + 1.41i·41^-s − 2.82i·43^-s − 6·47^-s − 10.9·49^-s + 8.48i·53^-s + ⋯

L(s) = 1

− 1.60i·7^-s + 1.20·11^-s + 1.66·13^-s − 1.02i·17^-s − 0.648i·19^-s + 1.25·23^-s + 25^-s + 1.57i·29^-s + 0.762i·31^-s + 0.986·37^-s + 0.220i·41^-s − 0.431i·43^-s − 0.875·47^-s − 1.57·49^-s + 1.16i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (4607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$2.526885928$
$L(\frac12)$	$\approx$	$2.526885928$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 - 5T^{2} $
	7	$ 1 + 4.24iT - 7T^{2} $
	11	$ 1 - 4T + 11T^{2} $
	13	$ 1 - 6T + 13T^{2} $
	17	$ 1 + 4.24iT - 17T^{2} $
	19	$ 1 + 2.82iT - 19T^{2} $
	23	$ 1 - 6T + 23T^{2} $
	29	$ 1 - 8.48iT - 29T^{2} $
	31	$ 1 - 4.24iT - 31T^{2} $
	37	$ 1 - 6T + 37T^{2} $
	41	$ 1 - 1.41iT - 41T^{2} $
	43	$ 1 + 2.82iT - 43T^{2} $
	47	$ 1 + 6T + 47T^{2} $
	53	$ 1 - 8.48iT - 53T^{2} $
	59	$ 1 + 4T + 59T^{2} $
	61	$ 1 + 6T + 61T^{2} $
	67	$ 1 + 11.3iT - 67T^{2} $
	71	$ 1 + 6T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 - 4.24iT - 79T^{2} $
	83	$ 1 - 16T + 83T^{2} $
	89	$ 1 - 12.7iT - 89T^{2} $
	97	$ 1 + 12T + 97T^{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

−8.223543303797020442650510989758, −7.26526592824132413419224349051, −6.81820840865345812992420634453, −6.31789516549105829615505192077, −5.04714353377098611534712036165, −4.47904134786138972421143943427, −3.56895280629375255955337280643, −3.07887638483274040555976741703, −1.30213643363648727244230431575, −0.939108887495370187249818808483, 1.14113503801256666518497202426, 2.01211967997663225720787154190, 3.08029807235012442454414782029, 3.83694027102871392400673482441, 4.68524519266277870022788608504, 5.85585439090744997395176792100, 6.05814012643584915620117160283, 6.72694023525528348780547049187, 7.989651474281278799881134277778, 8.484015295899434602061221337538

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(2.526885928\)
\(L(\frac12)\)	\(\approx\)	\(2.526885928\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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