LMFDB - L-function 2-1344-28.27-c3-0-7

Dirichlet series

L(s) = 1

− 3·3^-s + 4.57i·5^-s + (−2.93 − 18.2i)7^-s + 9·9^-s − 26.0i·11^-s + 75.0i·13^-s − 13.7i·15^-s − 115. i·17^-s − 119.·19^-s + (8.79 + 54.8i)21^-s + 61.4i·23^-s + 104.·25^-s − 27·27^-s + 71.9·29^-s − 231.·31^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.409i·5^-s + (−0.158 − 0.987i)7^-s + 0.333·9^-s − 0.713i·11^-s + 1.60i·13^-s − 0.236i·15^-s − 1.65i·17^-s − 1.43·19^-s + (0.0913 + 0.570i)21^-s + 0.556i·23^-s + 0.832·25^-s − 0.192·27^-s + 0.460·29^-s − 1.34·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$-0.158 - 0.987i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (895, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ -0.158 - 0.987i)$

Particular Values

$L(2)$	$\approx$	$0.6794404717$
$L(\frac12)$	$\approx$	$0.6794404717$
$L(\frac{5}{2})$		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 + 3T $
	7	$ 1 + (2.93 + 18.2i)T $
good	5	$ 1 - 4.57iT - 125T^{2} $
	11	$ 1 + 26.0iT - 1.33e3T^{2} $
	13	$ 1 - 75.0iT - 2.19e3T^{2} $
	17	$ 1 + 115. iT - 4.91e3T^{2} $
	19	$ 1 + 119.T + 6.85e3T^{2} $
	23	$ 1 - 61.4iT - 1.21e4T^{2} $
	29	$ 1 - 71.9T + 2.43e4T^{2} $
	31	$ 1 + 231.T + 2.97e4T^{2} $
	37	$ 1 + 13.3T + 5.06e4T^{2} $
	41	$ 1 - 144. iT - 6.89e4T^{2} $
	43	$ 1 + 288. iT - 7.95e4T^{2} $
	47	$ 1 - 343.T + 1.03e5T^{2} $
	53	$ 1 + 142.T + 1.48e5T^{2} $
	59	$ 1 - 403.T + 2.05e5T^{2} $
	61	$ 1 - 21.7iT - 2.26e5T^{2} $
	67	$ 1 + 598. iT - 3.00e5T^{2} $
	71	$ 1 + 589. iT - 3.57e5T^{2} $
	73	$ 1 - 6.85iT - 3.89e5T^{2} $
	79	$ 1 - 972. iT - 4.93e5T^{2} $
	83	$ 1 + 429.T + 5.71e5T^{2} $
	89	$ 1 - 1.08e3iT - 7.04e5T^{2} $
	97	$ 1 - 1.03e3iT - 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

−9.446303779656746194915904480444, −8.832328244818433901649950449685, −7.58362307330200409271337971376, −6.85826310272241630417152605262, −6.44631580397859082145826927226, −5.22253906781643909589133107053, −4.35483914532479132331172908097, −3.52900921713170939019034740828, −2.23275500569780386416891861008, −0.897563294278283776566939088296, 0.20596840664925417322307175055, 1.61239056880232223272147339338, 2.67879808269863299968313824402, 3.96374580286800835265693624330, 4.91182661375529837280045528002, 5.74651245629456402767906131818, 6.29448772985898612205911366528, 7.38531259388498535685611401087, 8.459448862333784309774839709078, 8.764140755395897586598202951020

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6794404717\)
\(L(\frac12)\)	\(\approx\)	\(0.6794404717\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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