LMFDB - L-function 2-1344-56.27-c3-0-58

Dirichlet series

L(s) = 1

+ 3i·3^-s − 1.64·5^-s + (−15.2 + 10.4i)7^-s − 9·9^-s − 20.3·11^-s − 13.0·13^-s − 4.92i·15^-s + 23.9i·17^-s + 87.7i·19^-s + (−31.4 − 45.8i)21^-s + 73.6i·23^-s − 122.·25^-s − 27i·27^-s + 58.9i·29^-s + 124.·31^-s + ⋯

L(s) = 1

+ 0.577i·3^-s − 0.146·5^-s + (−0.824 + 0.565i)7^-s − 0.333·9^-s − 0.557·11^-s − 0.279·13^-s − 0.0847i·15^-s + 0.341i·17^-s + 1.05i·19^-s + (−0.326 − 0.476i)21^-s + 0.667i·23^-s − 0.978·25^-s − 0.192i·27^-s + 0.377i·29^-s + 0.723·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.332 + 0.943i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.332 + 0.943i)$

Particular Values

$L(2)$	$\approx$	$0.3237040519$
$L(\frac12)$	$\approx$	$0.3237040519$
$L(\frac{5}{2})$		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 + (15.2 - 10.4i)T $
good	5	$ 1 + 1.64T + 125T^{2} $
	11	$ 1 + 20.3T + 1.33e3T^{2} $
	13	$ 1 + 13.0T + 2.19e3T^{2} $
	17	$ 1 - 23.9iT - 4.91e3T^{2} $
	19	$ 1 - 87.7iT - 6.85e3T^{2} $
	23	$ 1 - 73.6iT - 1.21e4T^{2} $
	29	$ 1 - 58.9iT - 2.43e4T^{2} $
	31	$ 1 - 124.T + 2.97e4T^{2} $
	37	$ 1 + 56.5iT - 5.06e4T^{2} $
	41	$ 1 - 135. iT - 6.89e4T^{2} $
	43	$ 1 + 259.T + 7.95e4T^{2} $
	47	$ 1 + 217.T + 1.03e5T^{2} $
	53	$ 1 + 529. iT - 1.48e5T^{2} $
	59	$ 1 - 685. iT - 2.05e5T^{2} $
	61	$ 1 + 149.T + 2.26e5T^{2} $
	67	$ 1 - 409.T + 3.00e5T^{2} $
	71	$ 1 + 885. iT - 3.57e5T^{2} $
	73	$ 1 + 269. iT - 3.89e5T^{2} $
	79	$ 1 + 902. iT - 4.93e5T^{2} $
	83	$ 1 - 623. iT - 5.71e5T^{2} $
	89	$ 1 + 986. iT - 7.04e5T^{2} $
	97	$ 1 + 179. iT - 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

−9.192606585917360753694953053820, −8.316669282568678736496220336336, −7.59853198500983749524350100053, −6.43560641938394770982174934491, −5.74266320942497302173363097590, −4.90182512124760573428540625107, −3.76604809029886815057413059456, −3.06072713080073834821970986089, −1.87368830559007105494122608512, −0.096365748724189139892515272173, 0.78733597964001700386792006559, 2.29920443691057837798301087458, 3.13684202429515561946585230033, 4.26782711119306582079454191723, 5.24231057751442429649960137323, 6.32534770991053083359436629052, 6.92959917227792282846648065993, 7.70272820083288144424598264408, 8.468154030288431088015639917420, 9.490710415081382810663378495513

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.3237040519\)
\(L(\frac12)\)	\(\approx\)	\(0.3237040519\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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