LMFDB - L-function 2-1344-56.27-c3-0-91

Dirichlet series

L(s) = 1

− 3i·3^-s − 12.8·5^-s + (−6.32 − 17.4i)7^-s − 9·9^-s − 25.0·11^-s + 64.1·13^-s + 38.4i·15^-s − 77.4i·17^-s − 49.9i·19^-s + (−52.2 + 18.9i)21^-s − 80.5i·23^-s + 39.4·25^-s + 27i·27^-s − 159. i·29^-s − 96.5·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.14·5^-s + (−0.341 − 0.939i)7^-s − 0.333·9^-s − 0.686·11^-s + 1.36·13^-s + 0.662i·15^-s − 1.10i·17^-s − 0.603i·19^-s + (−0.542 + 0.197i)21^-s − 0.730i·23^-s + 0.315·25^-s + 0.192i·27^-s − 1.01i·29^-s − 0.559·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$-0.573 - 0.819i$
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:	$(2,\ 1344,\ (\ :3/2),\ -0.573 - 0.819i)$

Particular Values

$L(2)$	$\approx$	$0.6079198789$
$L(\frac12)$	$\approx$	$0.6079198789$
$L(\frac{5}{2})$		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 + (6.32 + 17.4i)T $
good	5	$ 1 + 12.8T + 125T^{2} $
	11	$ 1 + 25.0T + 1.33e3T^{2} $
	13	$ 1 - 64.1T + 2.19e3T^{2} $
	17	$ 1 + 77.4iT - 4.91e3T^{2} $
	19	$ 1 + 49.9iT - 6.85e3T^{2} $
	23	$ 1 + 80.5iT - 1.21e4T^{2} $
	29	$ 1 + 159. iT - 2.43e4T^{2} $
	31	$ 1 + 96.5T + 2.97e4T^{2} $
	37	$ 1 + 274. iT - 5.06e4T^{2} $
	41	$ 1 + 299. iT - 6.89e4T^{2} $
	43	$ 1 - 385.T + 7.95e4T^{2} $
	47	$ 1 + 418.T + 1.03e5T^{2} $
	53	$ 1 + 665. iT - 1.48e5T^{2} $
	59	$ 1 - 445. iT - 2.05e5T^{2} $
	61	$ 1 - 599.T + 2.26e5T^{2} $
	67	$ 1 + 675.T + 3.00e5T^{2} $
	71	$ 1 + 877. iT - 3.57e5T^{2} $
	73	$ 1 - 696. iT - 3.89e5T^{2} $
	79	$ 1 - 1.24e3iT - 4.93e5T^{2} $
	83	$ 1 + 238. iT - 5.71e5T^{2} $
	89	$ 1 - 743. iT - 7.04e5T^{2} $
	97	$ 1 - 1.55e3iT - 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

−8.512591864636519477469357743534, −7.77854985779622623314133806035, −7.21500722350856912381642820072, −6.46048255547380110812546679208, −5.35946296582097279657708531495, −4.20365993960764201164816772317, −3.56252393914999107457760957147, −2.43381136359881596265094381135, −0.814040568764020112899134334853, −0.19251003687557528203801272174, 1.52964580881445982186614903161, 3.08800688440542811384518475357, 3.63632777637254086160464050399, 4.60007834473215020187677260412, 5.68173074071985907577351719139, 6.26698163486301040574582634006, 7.56182936232505502393644615843, 8.310562583362976621523940232139, 8.773118727757241061451329307839, 9.757138939233557364413197367204

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6079198789\)
\(L(\frac12)\)	\(\approx\)	\(0.6079198789\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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