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

Dirichlet series

L(s) = 1

− 3i·3^-s + 13.9i·5^-s − 7·7^-s − 9·9^-s + 51.5i·11^-s − 30.7i·13^-s + 41.9·15^-s + 0.246·17^-s − 2.72i·19^-s + 21i·21^-s − 14.5·23^-s − 70.6·25^-s + 27i·27^-s + 167. i·29^-s − 198.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.25i·5^-s − 0.377·7^-s − 0.333·9^-s + 1.41i·11^-s − 0.656i·13^-s + 0.722·15^-s + 0.00351·17^-s − 0.0328i·19^-s + 0.218i·21^-s − 0.131·23^-s − 0.565·25^-s + 0.192i·27^-s + 1.06i·29^-s − 1.15·31^-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

$L(2)$	$\approx$	$0.3254635803$
$L(\frac12)$	$\approx$	$0.3254635803$
$L(\frac{5}{2})$		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 - 13.9iT - 125T^{2} $
	11	$ 1 - 51.5iT - 1.33e3T^{2} $
	13	$ 1 + 30.7iT - 2.19e3T^{2} $
	17	$ 1 - 0.246T + 4.91e3T^{2} $
	19	$ 1 + 2.72iT - 6.85e3T^{2} $
	23	$ 1 + 14.5T + 1.21e4T^{2} $
	29	$ 1 - 167. iT - 2.43e4T^{2} $
	31	$ 1 + 198.T + 2.97e4T^{2} $
	37	$ 1 - 321. iT - 5.06e4T^{2} $
	41	$ 1 - 448.T + 6.89e4T^{2} $
	43	$ 1 + 86.5iT - 7.95e4T^{2} $
	47	$ 1 - 33.5T + 1.03e5T^{2} $
	53	$ 1 - 299. iT - 1.48e5T^{2} $
	59	$ 1 + 197. iT - 2.05e5T^{2} $
	61	$ 1 + 476. iT - 2.26e5T^{2} $
	67	$ 1 + 824. iT - 3.00e5T^{2} $
	71	$ 1 + 503.T + 3.57e5T^{2} $
	73	$ 1 + 1.12e3T + 3.89e5T^{2} $
	79	$ 1 + 684.T + 4.93e5T^{2} $
	83	$ 1 + 1.11e3iT - 5.71e5T^{2} $
	89	$ 1 + 643.T + 7.04e5T^{2} $
	97	$ 1 + 168.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

−9.786097509421701722846633749259, −8.910920802367583573421611593776, −7.70650134428805395082456695685, −7.24855603814596743502571400047, −6.57840317047561764948075979012, −5.73892152901026107532465554798, −4.60255898740507404918495386439, −3.36374027338662708538886580141, −2.62959739924786553069964056503, −1.57188434768747104303666330144, 0.079125861450166992614391754520, 1.10685537704800186773328598123, 2.57085508350562689835606977327, 3.81528374718732937432182893181, 4.38910232928922311626668366634, 5.58283266253833561487421681134, 5.93014759691207844223045514746, 7.25655577958172905401381516059, 8.272784893155397201163929956689, 8.952937245542303118589558187777

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.3254635803\)
\(L(\frac12)\)	\(\approx\)	\(0.3254635803\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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