LMFDB - L-function 2-1344-8.5-c3-0-71

Dirichlet series

L(s) = 1

− 3i·3^-s − 17.4i·5^-s − 7·7^-s − 9·9^-s − 37.3i·11^-s − 4.32i·13^-s − 52.4·15^-s − 0.453·17^-s + 58.2i·19^-s + 21i·21^-s − 181.·23^-s − 180.·25^-s + 27i·27^-s + 255. i·29^-s − 30.0·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.56i·5^-s − 0.377·7^-s − 0.333·9^-s − 1.02i·11^-s − 0.0922i·13^-s − 0.902·15^-s − 0.00646·17^-s + 0.703i·19^-s + 0.218i·21^-s − 1.64·23^-s − 1.44·25^-s + 0.192i·27^-s + 1.63i·29^-s − 0.174·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.258 - 0.965i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (673, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.258 - 0.965i)$

Particular Values

$L(2)$	$\approx$	$0.02959055015$
$L(\frac12)$	$\approx$	$0.02959055015$
$L(\frac{5}{2})$		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 + 7T $
good	5	$ 1 + 17.4iT - 125T^{2} $
	11	$ 1 + 37.3iT - 1.33e3T^{2} $
	13	$ 1 + 4.32iT - 2.19e3T^{2} $
	17	$ 1 + 0.453T + 4.91e3T^{2} $
	19	$ 1 - 58.2iT - 6.85e3T^{2} $
	23	$ 1 + 181.T + 1.21e4T^{2} $
	29	$ 1 - 255. iT - 2.43e4T^{2} $
	31	$ 1 + 30.0T + 2.97e4T^{2} $
	37	$ 1 + 135. iT - 5.06e4T^{2} $
	41	$ 1 - 30.5T + 6.89e4T^{2} $
	43	$ 1 + 423. iT - 7.95e4T^{2} $
	47	$ 1 + 28.9T + 1.03e5T^{2} $
	53	$ 1 - 117. iT - 1.48e5T^{2} $
	59	$ 1 - 447. iT - 2.05e5T^{2} $
	61	$ 1 + 22.4iT - 2.26e5T^{2} $
	67	$ 1 + 246. iT - 3.00e5T^{2} $
	71	$ 1 + 56.3T + 3.57e5T^{2} $
	73	$ 1 - 268.T + 3.89e5T^{2} $
	79	$ 1 + 964.T + 4.93e5T^{2} $
	83	$ 1 - 1.28e3iT - 5.71e5T^{2} $
	89	$ 1 + 269.T + 7.04e5T^{2} $
	97	$ 1 - 968.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

−8.567120837055729079420703674417, −8.010701780588957299575855800805, −7.00663179673175920930025190912, −5.83393512540789426545387235199, −5.50785559000281530549608843432, −4.28298750802196565725532840757, −3.37486957653142529243732073201, −1.93933882132734609036378176002, −0.974772637542653213092937570078, −0.00748666311283748220089030351, 2.08471113554068051305652038149, 2.86919689825519359758983013658, 3.84916768673417276573899933587, 4.65232447806296701956657720022, 5.97788948536032562297192303431, 6.53760136212469440867174435793, 7.39644134592052017009556876569, 8.130015076175190581462149691323, 9.431433420325873424692589311280, 9.945128952365925311163240243982

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.02959055015\)
\(L(\frac12)\)	\(\approx\)	\(0.02959055015\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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