LMFDB - L-function 2-1344-56.27-c3-0-36

Dirichlet series

L(s) = 1

− 3i·3^-s − 10.4·5^-s + (−9.40 + 15.9i)7^-s − 9·9^-s − 51.5·11^-s − 46.4·13^-s + 31.2i·15^-s + 99.1i·17^-s − 12.5i·19^-s + (47.8 + 28.2i)21^-s + 129. i·23^-s − 16.2·25^-s + 27i·27^-s + 207. i·29^-s + 139.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.932·5^-s + (−0.507 + 0.861i)7^-s − 0.333·9^-s − 1.41·11^-s − 0.990·13^-s + 0.538i·15^-s + 1.41i·17^-s − 0.151i·19^-s + (0.497 + 0.293i)21^-s + 1.17i·23^-s − 0.130·25^-s + 0.192i·27^-s + 1.32i·29^-s + 0.808·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.249 + 0.968i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.249 + 0.968i)$

Particular Values

$L(2)$	$\approx$	$0.2492406243$
$L(\frac12)$	$\approx$	$0.2492406243$
$L(\frac{5}{2})$		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 + (9.40 - 15.9i)T $
good	5	$ 1 + 10.4T + 125T^{2} $
	11	$ 1 + 51.5T + 1.33e3T^{2} $
	13	$ 1 + 46.4T + 2.19e3T^{2} $
	17	$ 1 - 99.1iT - 4.91e3T^{2} $
	19	$ 1 + 12.5iT - 6.85e3T^{2} $
	23	$ 1 - 129. iT - 1.21e4T^{2} $
	29	$ 1 - 207. iT - 2.43e4T^{2} $
	31	$ 1 - 139.T + 2.97e4T^{2} $
	37	$ 1 + 152. iT - 5.06e4T^{2} $
	41	$ 1 + 317. iT - 6.89e4T^{2} $
	43	$ 1 + 366.T + 7.95e4T^{2} $
	47	$ 1 + 103.T + 1.03e5T^{2} $
	53	$ 1 + 24.2iT - 1.48e5T^{2} $
	59	$ 1 + 339. iT - 2.05e5T^{2} $
	61	$ 1 + 25.3T + 2.26e5T^{2} $
	67	$ 1 + 660.T + 3.00e5T^{2} $
	71	$ 1 - 0.771iT - 3.57e5T^{2} $
	73	$ 1 - 342. iT - 3.89e5T^{2} $
	79	$ 1 - 951. iT - 4.93e5T^{2} $
	83	$ 1 - 244. iT - 5.71e5T^{2} $
	89	$ 1 + 1.17e3iT - 7.04e5T^{2} $
	97	$ 1 - 1.40e3iT - 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

−8.876695183029239857608060023964, −8.131582722610569701894251295709, −7.56672452159757200799414382568, −6.73569446986210133014121189737, −5.67047758425867481095954088373, −5.03952157658306101209638931006, −3.70157792530640569255956176293, −2.83115398329414084998857800202, −1.83646190098503615007004411571, −0.11501414407927151933200118667, 0.48314670357351972915367667130, 2.58962643401937246925774132049, 3.25793741036093883774280651725, 4.56232242199402206855724071633, 4.75898295356136210148782500173, 6.12576679886626826068462696521, 7.18520276847863185581616954569, 7.76083213733848966687618639703, 8.449344323339202833981949963180, 9.815736249676991576316208321105

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.2492406243\)
\(L(\frac12)\)	\(\approx\)	\(0.2492406243\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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