LMFDB - L-function 2-1344-8.5-c3-0-54

Dirichlet series

L(s) = 1

− 3i·3^-s + 6.24i·5^-s + 7·7^-s − 9·9^-s − 63.4i·11^-s + 82.2i·13^-s + 18.7·15^-s + 75.4·17^-s − 125. i·19^-s − 21i·21^-s − 155.·23^-s + 85.9·25^-s + 27i·27^-s − 56.2i·29^-s − 159.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.558i·5^-s + 0.377·7^-s − 0.333·9^-s − 1.73i·11^-s + 1.75i·13^-s + 0.322·15^-s + 1.07·17^-s − 1.51i·19^-s − 0.218i·21^-s − 1.40·23^-s + 0.687·25^-s + 0.192i·27^-s − 0.360i·29^-s − 0.922·31^-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

$L(2)$	$\approx$	$1.666544366$
$L(\frac12)$	$\approx$	$1.666544366$
$L(\frac{5}{2})$		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 - 6.24iT - 125T^{2} $
	11	$ 1 + 63.4iT - 1.33e3T^{2} $
	13	$ 1 - 82.2iT - 2.19e3T^{2} $
	17	$ 1 - 75.4T + 4.91e3T^{2} $
	19	$ 1 + 125. iT - 6.85e3T^{2} $
	23	$ 1 + 155.T + 1.21e4T^{2} $
	29	$ 1 + 56.2iT - 2.43e4T^{2} $
	31	$ 1 + 159.T + 2.97e4T^{2} $
	37	$ 1 - 197. iT - 5.06e4T^{2} $
	41	$ 1 - 137.T + 6.89e4T^{2} $
	43	$ 1 - 295. iT - 7.95e4T^{2} $
	47	$ 1 - 186.T + 1.03e5T^{2} $
	53	$ 1 + 409. iT - 1.48e5T^{2} $
	59	$ 1 + 311. iT - 2.05e5T^{2} $
	61	$ 1 + 168. iT - 2.26e5T^{2} $
	67	$ 1 + 563. iT - 3.00e5T^{2} $
	71	$ 1 - 282.T + 3.57e5T^{2} $
	73	$ 1 - 250.T + 3.89e5T^{2} $
	79	$ 1 + 948.T + 4.93e5T^{2} $
	83	$ 1 + 1.28e3iT - 5.71e5T^{2} $
	89	$ 1 + 655.T + 7.04e5T^{2} $
	97	$ 1 - 706.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

−8.886448970314811891108379592371, −8.202637329371032980008719360022, −7.32767791373316477945630838526, −6.51783777479047833146744663962, −5.91752144365266278466409273661, −4.79981676705828627150007009896, −3.65296076746391529365075977024, −2.73590567119428233334116346455, −1.62377485601993510118963131349, −0.41285418062095750269134669372, 1.11139246096518543949350364191, 2.27094437162044015640937187143, 3.58958590384448169747040149904, 4.35370392222941797132268787179, 5.41056037198173125144187327553, 5.73477185548854364371187721424, 7.34772383715905777276043000272, 7.83116249916194385406163212268, 8.652488861276068065010491702532, 9.657840405660110157897824483678

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.666544366\)
\(L(\frac12)\)	\(\approx\)	\(1.666544366\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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