LMFDB - L-function 2-1344-8.5-c3-0-35

Dirichlet series

L(s) = 1

− 3i·3^-s − 7.86i·5^-s − 7·7^-s − 9·9^-s − 31.0i·11^-s + 48.9i·13^-s − 23.5·15^-s − 43.0·17^-s + 110. i·19^-s + 21i·21^-s + 170.·23^-s + 63.1·25^-s + 27i·27^-s + 78.6i·29^-s − 48.8·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.703i·5^-s − 0.377·7^-s − 0.333·9^-s − 0.852i·11^-s + 1.04i·13^-s − 0.406·15^-s − 0.614·17^-s + 1.32i·19^-s + 0.218i·21^-s + 1.54·23^-s + 0.505·25^-s + 0.192i·27^-s + 0.503i·29^-s − 0.283·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.707 + 0.707i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.876261888$
$L(\frac12)$	$\approx$	$1.876261888$
$L(\frac{5}{2})$		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 + 7.86iT - 125T^{2} $
	11	$ 1 + 31.0iT - 1.33e3T^{2} $
	13	$ 1 - 48.9iT - 2.19e3T^{2} $
	17	$ 1 + 43.0T + 4.91e3T^{2} $
	19	$ 1 - 110. iT - 6.85e3T^{2} $
	23	$ 1 - 170.T + 1.21e4T^{2} $
	29	$ 1 - 78.6iT - 2.43e4T^{2} $
	31	$ 1 + 48.8T + 2.97e4T^{2} $
	37	$ 1 + 6.95iT - 5.06e4T^{2} $
	41	$ 1 - 339.T + 6.89e4T^{2} $
	43	$ 1 - 222. iT - 7.95e4T^{2} $
	47	$ 1 - 314.T + 1.03e5T^{2} $
	53	$ 1 + 487. iT - 1.48e5T^{2} $
	59	$ 1 + 500. iT - 2.05e5T^{2} $
	61	$ 1 - 29.6iT - 2.26e5T^{2} $
	67	$ 1 - 380. iT - 3.00e5T^{2} $
	71	$ 1 + 12.5T + 3.57e5T^{2} $
	73	$ 1 + 857.T + 3.89e5T^{2} $
	79	$ 1 - 799.T + 4.93e5T^{2} $
	83	$ 1 - 178. iT - 5.71e5T^{2} $
	89	$ 1 + 1.02e3T + 7.04e5T^{2} $
	97	$ 1 + 360.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

−8.915967836256922411653666754168, −8.542949535783831745473208996494, −7.46550655151042786792487995902, −6.67308647723868552231219723580, −5.90825825312262930155866593815, −4.98709542500609048422288682609, −3.95865436252450690146902397095, −2.90166834669383786260111222801, −1.65321391496088608144367557866, −0.69726987852685125227083853783, 0.68478191193433452821901419265, 2.49833661136670240110265349502, 3.06200904632881474732560586906, 4.26032224424791044973091133288, 5.04819573117454376589644040683, 6.03227015589732954193586741631, 7.01833437168976903568284483477, 7.48861313509563113799468751847, 8.835676577529736455595317832453, 9.266795796157296723981775011700

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.876261888\)
\(L(\frac12)\)	\(\approx\)	\(1.876261888\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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