LMFDB - L-function 2-1344-56.27-c3-0-73

Dirichlet series

L(s) = 1

− 3i·3^-s + 20.7·5^-s + (11.8 − 14.2i)7^-s − 9·9^-s + 39.2·11^-s − 36.2·13^-s − 62.1i·15^-s + 92.2i·17^-s − 87.5i·19^-s + (−42.7 − 35.4i)21^-s + 100. i·23^-s + 303.·25^-s + 27i·27^-s + 36.2i·29^-s + 264.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.85·5^-s + (0.638 − 0.770i)7^-s − 0.333·9^-s + 1.07·11^-s − 0.773·13^-s − 1.06i·15^-s + 1.31i·17^-s − 1.05i·19^-s + (−0.444 − 0.368i)21^-s + 0.915i·23^-s + 2.42·25^-s + 0.192i·27^-s + 0.231i·29^-s + 1.53·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.578 + 0.815i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.578 + 0.815i)$

Particular Values

$L(2)$	$\approx$	$3.769084860$
$L(\frac12)$	$\approx$	$3.769084860$
$L(\frac{5}{2})$		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 + (-11.8 + 14.2i)T $
good	5	$ 1 - 20.7T + 125T^{2} $
	11	$ 1 - 39.2T + 1.33e3T^{2} $
	13	$ 1 + 36.2T + 2.19e3T^{2} $
	17	$ 1 - 92.2iT - 4.91e3T^{2} $
	19	$ 1 + 87.5iT - 6.85e3T^{2} $
	23	$ 1 - 100. iT - 1.21e4T^{2} $
	29	$ 1 - 36.2iT - 2.43e4T^{2} $
	31	$ 1 - 264.T + 2.97e4T^{2} $
	37	$ 1 + 229. iT - 5.06e4T^{2} $
	41	$ 1 + 337. iT - 6.89e4T^{2} $
	43	$ 1 - 368.T + 7.95e4T^{2} $
	47	$ 1 + 314.T + 1.03e5T^{2} $
	53	$ 1 + 108. iT - 1.48e5T^{2} $
	59	$ 1 - 185. iT - 2.05e5T^{2} $
	61	$ 1 + 163.T + 2.26e5T^{2} $
	67	$ 1 - 458.T + 3.00e5T^{2} $
	71	$ 1 - 584. iT - 3.57e5T^{2} $
	73	$ 1 - 1.09e3iT - 3.89e5T^{2} $
	79	$ 1 + 438. iT - 4.93e5T^{2} $
	83	$ 1 - 722. iT - 5.71e5T^{2} $
	89	$ 1 + 452. iT - 7.04e5T^{2} $
	97	$ 1 + 597. 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

−9.185377761910238024772721636970, −8.408208870525879789357346070195, −7.26533939615623562773187264773, −6.67005756118835134289328784799, −5.88133396522625747467145640780, −5.08076331722393805189276761769, −4.00840181036604916734686352236, −2.55149077981229095361476917067, −1.71209902423568012099946589579, −0.965337125363911928638868159859, 1.17374631734500170152142682964, 2.22012586847288549301858914558, 2.95526028885130660285498385980, 4.62324127081206352758985129872, 5.06302786870521896846341043051, 6.11122769170163184451055764703, 6.50503086350112381577585164750, 7.912396490202562387953814224423, 8.854710651429702548444672455506, 9.518093331290070342515016731252

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.769084860\)
\(L(\frac12)\)	\(\approx\)	\(3.769084860\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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