LMFDB - L-function 2-1344-56.27-c3-0-74

Dirichlet series

L(s) = 1

− 3i·3^-s + 3.15·5^-s + (6.11 + 17.4i)7^-s − 9·9^-s − 60.9·11^-s + 59.6·13^-s − 9.46i·15^-s − 21.4i·17^-s + 95.5i·19^-s + (52.4 − 18.3i)21^-s − 46.3i·23^-s − 115.·25^-s + 27i·27^-s − 107. i·29^-s − 94.2·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.282·5^-s + (0.329 + 0.944i)7^-s − 0.333·9^-s − 1.67·11^-s + 1.27·13^-s − 0.162i·15^-s − 0.305i·17^-s + 1.15i·19^-s + (0.545 − 0.190i)21^-s − 0.419i·23^-s − 0.920·25^-s + 0.192i·27^-s − 0.688i·29^-s − 0.546·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$-0.826 + 0.563i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (223, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ -0.826 + 0.563i)$

Particular Values

$L(2)$	$\approx$	$0.7145302467$
$L(\frac12)$	$\approx$	$0.7145302467$
$L(\frac{5}{2})$		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 + 3iT $
	7	$ 1 + (-6.11 - 17.4i)T $
good	5	$ 1 - 3.15T + 125T^{2} $
	11	$ 1 + 60.9T + 1.33e3T^{2} $
	13	$ 1 - 59.6T + 2.19e3T^{2} $
	17	$ 1 + 21.4iT - 4.91e3T^{2} $
	19	$ 1 - 95.5iT - 6.85e3T^{2} $
	23	$ 1 + 46.3iT - 1.21e4T^{2} $
	29	$ 1 + 107. iT - 2.43e4T^{2} $
	31	$ 1 + 94.2T + 2.97e4T^{2} $
	37	$ 1 - 131. iT - 5.06e4T^{2} $
	41	$ 1 + 283. iT - 6.89e4T^{2} $
	43	$ 1 - 373.T + 7.95e4T^{2} $
	47	$ 1 + 136.T + 1.03e5T^{2} $
	53	$ 1 + 298. iT - 1.48e5T^{2} $
	59	$ 1 + 468. iT - 2.05e5T^{2} $
	61	$ 1 + 563.T + 2.26e5T^{2} $
	67	$ 1 + 160.T + 3.00e5T^{2} $
	71	$ 1 + 409. iT - 3.57e5T^{2} $
	73	$ 1 + 930. iT - 3.89e5T^{2} $
	79	$ 1 - 442. iT - 4.93e5T^{2} $
	83	$ 1 - 190. iT - 5.71e5T^{2} $
	89	$ 1 + 829. iT - 7.04e5T^{2} $
	97	$ 1 + 1.03e3iT - 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

−8.717800651453941378341952243309, −8.086190804464133954374710809669, −7.51301430350633567298130853981, −6.12236738443386104220656447130, −5.80775671901327796793089516157, −4.88525342084439002706489456329, −3.51192023966584392973644770421, −2.44164935920000651352771471980, −1.67851807458978102774278025418, −0.16298849599597011194340656693, 1.17713449151786176452246168294, 2.54471307064216886777821336343, 3.58789949351787440965121451963, 4.47634339523785329871217376444, 5.34463724123891224026319210319, 6.11181211652418748715132111197, 7.29719134232327530219463671754, 7.927904666125376670654278091052, 8.800905390494616370093750902197, 9.621343580276810259981703025815

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7145302467\)
\(L(\frac12)\)	\(\approx\)	\(0.7145302467\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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