LMFDB - L-function 2-1344-8.5-c3-0-9

Dirichlet series

L(s) = 1

− 3i·3^-s − 1.12i·5^-s − 7·7^-s − 9·9^-s + 16.9i·11^-s − 11.6i·13^-s − 3.37·15^-s − 69.2·17^-s − 87.3i·19^-s + 21i·21^-s + 49.5·23^-s + 123.·25^-s + 27i·27^-s − 178. i·29^-s − 147.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.100i·5^-s − 0.377·7^-s − 0.333·9^-s + 0.464i·11^-s − 0.248i·13^-s − 0.0581·15^-s − 0.988·17^-s − 1.05i·19^-s + 0.218i·21^-s + 0.449·23^-s + 0.989·25^-s + 0.192i·27^-s − 1.14i·29^-s − 0.853·31^-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

$L(2)$	$\approx$	$0.8520128977$
$L(\frac12)$	$\approx$	$0.8520128977$
$L(\frac{5}{2})$		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 + 1.12iT - 125T^{2} $
	11	$ 1 - 16.9iT - 1.33e3T^{2} $
	13	$ 1 + 11.6iT - 2.19e3T^{2} $
	17	$ 1 + 69.2T + 4.91e3T^{2} $
	19	$ 1 + 87.3iT - 6.85e3T^{2} $
	23	$ 1 - 49.5T + 1.21e4T^{2} $
	29	$ 1 + 178. iT - 2.43e4T^{2} $
	31	$ 1 + 147.T + 2.97e4T^{2} $
	37	$ 1 - 241. iT - 5.06e4T^{2} $
	41	$ 1 + 61.3T + 6.89e4T^{2} $
	43	$ 1 - 283. iT - 7.95e4T^{2} $
	47	$ 1 + 300.T + 1.03e5T^{2} $
	53	$ 1 + 44.4iT - 1.48e5T^{2} $
	59	$ 1 - 260. iT - 2.05e5T^{2} $
	61	$ 1 + 151. iT - 2.26e5T^{2} $
	67	$ 1 - 635. iT - 3.00e5T^{2} $
	71	$ 1 + 51.2T + 3.57e5T^{2} $
	73	$ 1 + 346.T + 3.89e5T^{2} $
	79	$ 1 - 173.T + 4.93e5T^{2} $
	83	$ 1 - 524. iT - 5.71e5T^{2} $
	89	$ 1 - 599.T + 7.04e5T^{2} $
	97	$ 1 - 468.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

−9.289513244622325722402838749788, −8.644400832743043211911687659532, −7.72245805469529039606788436735, −6.86601828311155753366835596872, −6.36369012658464718888121763078, −5.18844030432890347772825770695, −4.39976263714096479513817785589, −3.10235240668953745802608901823, −2.24178135814874820603644643874, −0.963216179892262018136419821998, 0.22155911915149255873391205329, 1.78212723646009715168548040445, 3.06356754813522507480444247348, 3.80736345856452194043947652034, 4.83169489320879065257026206349, 5.69190199013853475202272678270, 6.59089318698202290529945587998, 7.36795859784348258621772858422, 8.554744431311068848001487544530, 9.005450235182383632260706933049

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8520128977\)
\(L(\frac12)\)	\(\approx\)	\(0.8520128977\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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