LMFDB - L-function 2-1344-8.5-c3-0-11

Dirichlet series

L(s) = 1

− 3i·3^-s + 20.9i·5^-s + 7·7^-s − 9·9^-s + 10.4i·11^-s + 50.9i·13^-s + 62.9·15^-s + 19.1·17^-s + 24.3i·19^-s − 21i·21^-s − 101.·23^-s − 315.·25^-s + 27i·27^-s + 179. i·29^-s + 264.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.87i·5^-s + 0.377·7^-s − 0.333·9^-s + 0.285i·11^-s + 1.08i·13^-s + 1.08·15^-s + 0.273·17^-s + 0.294i·19^-s − 0.218i·21^-s − 0.921·23^-s − 2.52·25^-s + 0.192i·27^-s + 1.14i·29^-s + 1.53·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$-0.965 - 0.258i$
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:	$(2,\ 1344,\ (\ :3/2),\ -0.965 - 0.258i)$

Particular Values

$L(2)$	$\approx$	$1.314064499$
$L(\frac12)$	$\approx$	$1.314064499$
$L(\frac{5}{2})$		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 - 20.9iT - 125T^{2} $
	11	$ 1 - 10.4iT - 1.33e3T^{2} $
	13	$ 1 - 50.9iT - 2.19e3T^{2} $
	17	$ 1 - 19.1T + 4.91e3T^{2} $
	19	$ 1 - 24.3iT - 6.85e3T^{2} $
	23	$ 1 + 101.T + 1.21e4T^{2} $
	29	$ 1 - 179. iT - 2.43e4T^{2} $
	31	$ 1 - 264.T + 2.97e4T^{2} $
	37	$ 1 - 328. iT - 5.06e4T^{2} $
	41	$ 1 + 89.3T + 6.89e4T^{2} $
	43	$ 1 + 124. iT - 7.95e4T^{2} $
	47	$ 1 - 446.T + 1.03e5T^{2} $
	53	$ 1 + 384. iT - 1.48e5T^{2} $
	59	$ 1 + 94.7iT - 2.05e5T^{2} $
	61	$ 1 + 376. iT - 2.26e5T^{2} $
	67	$ 1 - 338. iT - 3.00e5T^{2} $
	71	$ 1 - 268.T + 3.57e5T^{2} $
	73	$ 1 + 634.T + 3.89e5T^{2} $
	79	$ 1 + 1.13e3T + 4.93e5T^{2} $
	83	$ 1 + 589. iT - 5.71e5T^{2} $
	89	$ 1 + 459.T + 7.04e5T^{2} $
	97	$ 1 + 1.36e3T + 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

−9.909606184805645664622972877310, −8.637751905748315882672320158440, −7.81462679126521972892391754591, −7.00147032198745217505224363490, −6.59784899116145753393792873319, −5.73613031703998603969997425848, −4.40134159487295828797425800803, −3.35853564766103899532741671276, −2.46220713057027118311031698305, −1.57168049260616645998613705864, 0.31343349092667232792238598057, 1.17134907863042365320050145636, 2.57136152229692493727408255893, 4.00724800744592738767765323990, 4.54561716992071688860950315535, 5.51166868550255767439213513808, 5.91218297254537756874692666136, 7.61705884350659687244334463938, 8.242073876462553641820725390964, 8.795404806943087043847012886234

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.314064499\)
\(L(\frac12)\)	\(\approx\)	\(1.314064499\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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