LMFDB - L-function 2-1344-56.27-c3-0-55

Dirichlet series

L(s) = 1

+ 3i·3^-s − 12.8·5^-s + (6.32 + 17.4i)7^-s − 9·9^-s + 25.0·11^-s + 64.1·13^-s − 38.4i·15^-s − 77.4i·17^-s + 49.9i·19^-s + (−52.2 + 18.9i)21^-s + 80.5i·23^-s + 39.4·25^-s − 27i·27^-s − 159. i·29^-s + 96.5·31^-s + ⋯

L(s) = 1

+ 0.577i·3^-s − 1.14·5^-s + (0.341 + 0.939i)7^-s − 0.333·9^-s + 0.686·11^-s + 1.36·13^-s − 0.662i·15^-s − 1.10i·17^-s + 0.603i·19^-s + (−0.542 + 0.197i)21^-s + 0.730i·23^-s + 0.315·25^-s − 0.192i·27^-s − 1.01i·29^-s + 0.559·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.819 - 0.573i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.819 - 0.573i)$

Particular Values

$L(2)$	$\approx$	$1.845137725$
$L(\frac12)$	$\approx$	$1.845137725$
$L(\frac{5}{2})$		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 + (-6.32 - 17.4i)T $
good	5	$ 1 + 12.8T + 125T^{2} $
	11	$ 1 - 25.0T + 1.33e3T^{2} $
	13	$ 1 - 64.1T + 2.19e3T^{2} $
	17	$ 1 + 77.4iT - 4.91e3T^{2} $
	19	$ 1 - 49.9iT - 6.85e3T^{2} $
	23	$ 1 - 80.5iT - 1.21e4T^{2} $
	29	$ 1 + 159. iT - 2.43e4T^{2} $
	31	$ 1 - 96.5T + 2.97e4T^{2} $
	37	$ 1 + 274. iT - 5.06e4T^{2} $
	41	$ 1 + 299. iT - 6.89e4T^{2} $
	43	$ 1 + 385.T + 7.95e4T^{2} $
	47	$ 1 - 418.T + 1.03e5T^{2} $
	53	$ 1 + 665. iT - 1.48e5T^{2} $
	59	$ 1 + 445. iT - 2.05e5T^{2} $
	61	$ 1 - 599.T + 2.26e5T^{2} $
	67	$ 1 - 675.T + 3.00e5T^{2} $
	71	$ 1 - 877. iT - 3.57e5T^{2} $
	73	$ 1 - 696. iT - 3.89e5T^{2} $
	79	$ 1 + 1.24e3iT - 4.93e5T^{2} $
	83	$ 1 - 238. iT - 5.71e5T^{2} $
	89	$ 1 - 743. iT - 7.04e5T^{2} $
	97	$ 1 - 1.55e3iT - 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

−9.206348513229047090374724667716, −8.511573450298802520021916351274, −7.939468254325984944774576004656, −6.88806643643404344313290170163, −5.86990883471926307741109311634, −5.10516272538238260751368388183, −3.94591967845853389350549316519, −3.54628152624926621414080032343, −2.16600871255323077968343416357, −0.67227302627022618757483171474, 0.75739727061452556738260725437, 1.52118811726926010135124149449, 3.20671798270406420303743442921, 3.96491869128738761767105621002, 4.68794735131738499831297452965, 6.14366379793183340822456049849, 6.75605170951896313322328476867, 7.57005939904568873849146554491, 8.370740302696201523883682661360, 8.737901611309333570090077642315

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.845137725\)
\(L(\frac12)\)	\(\approx\)	\(1.845137725\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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