LMFDB - L-function 2-1344-8.5-c3-0-58

Dirichlet series

L(s) = 1

− 3i·3^-s − 2.27i·5^-s − 7·7^-s − 9·9^-s + 1.28i·11^-s + 84.9i·13^-s − 6.81·15^-s + 49.4·17^-s + 2.27i·19^-s + 21i·21^-s − 96.7·23^-s + 119.·25^-s + 27i·27^-s − 216. i·29^-s + 108.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.203i·5^-s − 0.377·7^-s − 0.333·9^-s + 0.0353i·11^-s + 1.81i·13^-s − 0.117·15^-s + 0.705·17^-s + 0.0274i·19^-s + 0.218i·21^-s − 0.876·23^-s + 0.958·25^-s + 0.192i·27^-s − 1.38i·29^-s + 0.627·31^-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

$L(2)$	$\approx$	$0.6166829340$
$L(\frac12)$	$\approx$	$0.6166829340$
$L(\frac{5}{2})$		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 + 2.27iT - 125T^{2} $
	11	$ 1 - 1.28iT - 1.33e3T^{2} $
	13	$ 1 - 84.9iT - 2.19e3T^{2} $
	17	$ 1 - 49.4T + 4.91e3T^{2} $
	19	$ 1 - 2.27iT - 6.85e3T^{2} $
	23	$ 1 + 96.7T + 1.21e4T^{2} $
	29	$ 1 + 216. iT - 2.43e4T^{2} $
	31	$ 1 - 108.T + 2.97e4T^{2} $
	37	$ 1 + 92.2iT - 5.06e4T^{2} $
	41	$ 1 + 356.T + 6.89e4T^{2} $
	43	$ 1 + 500. iT - 7.95e4T^{2} $
	47	$ 1 + 104.T + 1.03e5T^{2} $
	53	$ 1 - 53.9iT - 1.48e5T^{2} $
	59	$ 1 + 395. iT - 2.05e5T^{2} $
	61	$ 1 + 76.1iT - 2.26e5T^{2} $
	67	$ 1 + 55.9iT - 3.00e5T^{2} $
	71	$ 1 + 1.00e3T + 3.57e5T^{2} $
	73	$ 1 + 149.T + 3.89e5T^{2} $
	79	$ 1 + 331.T + 4.93e5T^{2} $
	83	$ 1 - 729. iT - 5.71e5T^{2} $
	89	$ 1 + 7.31T + 7.04e5T^{2} $
	97	$ 1 + 1.76e3T + 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

−8.828708394141167404893678513540, −8.094022683651384913872453955621, −7.09979116642135157857700705442, −6.53337437351300543769540841849, −5.66285052975215386781091205819, −4.55888023906422804733817954529, −3.66839606206749603287206787195, −2.40743187558250558144410010823, −1.47959695002579496356242593646, −0.14816414887673409460087698057, 1.18309535980249316597013871461, 2.93102300052683017634725673612, 3.29346391718261913790356899822, 4.59180716022973633804322615933, 5.43946480461632503937184597446, 6.17622519425346051633929441773, 7.21872257287455585939637213617, 8.116546296406593020997610550723, 8.741134889407907641816499553763, 9.904936041496228927229809926534

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6166829340\)
\(L(\frac12)\)	\(\approx\)	\(0.6166829340\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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