LMFDB - L-function 2-624-12.11-c3-0-1

Dirichlet series

L(s) = 1

+ (2.22 − 4.69i)3^-s + 16.3i·5^-s + 31.0i·7^-s + (−17.0 − 20.9i)9^-s − 35.9·11^-s + 13·13^-s + (76.8 + 36.4i)15^-s + 61.5i·17^-s − 66.6i·19^-s + (145. + 69.1i)21^-s − 183.·23^-s − 142.·25^-s + (−136. + 33.3i)27^-s − 44.2i·29^-s − 291. i·31^-s + ⋯

L(s) = 1

+ (0.429 − 0.903i)3^-s + 1.46i·5^-s + 1.67i·7^-s + (−0.631 − 0.775i)9^-s − 0.986·11^-s + 0.277·13^-s + (1.32 + 0.628i)15^-s + 0.877i·17^-s − 0.804i·19^-s + (1.51 + 0.719i)21^-s − 1.66·23^-s − 1.14·25^-s + (−0.971 + 0.238i)27^-s − 0.283i·29^-s − 1.68i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0800i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$624$ = $2^{4} \cdot 3 \cdot 13$
Sign:	$-0.996 + 0.0800i$
Analytic conductor:	$36.8171$
Root analytic conductor:	$6.06771$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{624} (287, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 624,\ (\ :3/2),\ -0.996 + 0.0800i)$

Particular Values

$L(2)$	$\approx$	$0.3758407494$
$L(\frac12)$	$\approx$	$0.3758407494$
$L(\frac{5}{2})$		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 + (-2.22 + 4.69i)T $
	13	$ 1 - 13T $
good	5	$ 1 - 16.3iT - 125T^{2} $
	7	$ 1 - 31.0iT - 343T^{2} $
	11	$ 1 + 35.9T + 1.33e3T^{2} $
	17	$ 1 - 61.5iT - 4.91e3T^{2} $
	19	$ 1 + 66.6iT - 6.85e3T^{2} $
	23	$ 1 + 183.T + 1.21e4T^{2} $
	29	$ 1 + 44.2iT - 2.43e4T^{2} $
	31	$ 1 + 291. iT - 2.97e4T^{2} $
	37	$ 1 - 27.9T + 5.06e4T^{2} $
	41	$ 1 + 219. iT - 6.89e4T^{2} $
	43	$ 1 + 409. iT - 7.95e4T^{2} $
	47	$ 1 - 317.T + 1.03e5T^{2} $
	53	$ 1 + 123. iT - 1.48e5T^{2} $
	59	$ 1 - 100.T + 2.05e5T^{2} $
	61	$ 1 + 254.T + 2.26e5T^{2} $
	67	$ 1 - 193. iT - 3.00e5T^{2} $
	71	$ 1 + 109.T + 3.57e5T^{2} $
	73	$ 1 + 708.T + 3.89e5T^{2} $
	79	$ 1 - 1.01e3iT - 4.93e5T^{2} $
	83	$ 1 + 991.T + 5.71e5T^{2} $
	89	$ 1 - 1.37e3iT - 7.04e5T^{2} $
	97	$ 1 - 154.T + 9.12e5T^{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

−10.73393581023031034950382001861, −9.759256159392557995676762775292, −8.705862405941942872875354256132, −8.005290866152297459653929115211, −7.14686488966066869085767611885, −6.08530729306345318392333469214, −5.70815627441247596697649892490, −3.72252372122366533418821812427, −2.43950451328734705469608704360, −2.29113867117877126719773906365, 0.096656894373888303761048797847, 1.43265187457875051278079425321, 3.19862058878847399432878365909, 4.31069481867191384121196518158, 4.76916320994487103497442781752, 5.83884101329370886537414137615, 7.46022019895818217902290272670, 8.089055486778515083593772826087, 8.881648747677618286983903566662, 9.949954577956089506087950151789

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(L(2)\)	\(\approx\)	\(0.3758407494\)
\(L(\frac12)\)	\(\approx\)	\(0.3758407494\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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