LMFDB - L-function 2-1344-28.27-c3-0-6

Dirichlet series

L(s) = 1

+ 3·3^-s − 14.9i·5^-s + (−18.3 − 2.46i)7^-s + 9·9^-s + 21.4i·11^-s − 66.7i·13^-s − 44.8i·15^-s + 66.3i·17^-s − 126.·19^-s + (−55.0 − 7.39i)21^-s + 153. i·23^-s − 98.6·25^-s + 27·27^-s + 198.·29^-s − 273.·31^-s + ⋯

L(s) = 1

+ 0.577·3^-s − 1.33i·5^-s + (−0.991 − 0.133i)7^-s + 0.333·9^-s + 0.588i·11^-s − 1.42i·13^-s − 0.772i·15^-s + 0.946i·17^-s − 1.53·19^-s + (−0.572 − 0.0768i)21^-s + 1.39i·23^-s − 0.788·25^-s + 0.192·27^-s + 1.27·29^-s − 1.58·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.133 - 0.991i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.133 - 0.991i)$

Particular Values

$L(2)$	$\approx$	$0.8195174541$
$L(\frac12)$	$\approx$	$0.8195174541$
$L(\frac{5}{2})$		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 + (18.3 + 2.46i)T $
good	5	$ 1 + 14.9iT - 125T^{2} $
	11	$ 1 - 21.4iT - 1.33e3T^{2} $
	13	$ 1 + 66.7iT - 2.19e3T^{2} $
	17	$ 1 - 66.3iT - 4.91e3T^{2} $
	19	$ 1 + 126.T + 6.85e3T^{2} $
	23	$ 1 - 153. iT - 1.21e4T^{2} $
	29	$ 1 - 198.T + 2.43e4T^{2} $
	31	$ 1 + 273.T + 2.97e4T^{2} $
	37	$ 1 + 135.T + 5.06e4T^{2} $
	41	$ 1 + 89.7iT - 6.89e4T^{2} $
	43	$ 1 + 115. iT - 7.95e4T^{2} $
	47	$ 1 - 74.7T + 1.03e5T^{2} $
	53	$ 1 - 484.T + 1.48e5T^{2} $
	59	$ 1 + 410.T + 2.05e5T^{2} $
	61	$ 1 - 753. iT - 2.26e5T^{2} $
	67	$ 1 - 684. iT - 3.00e5T^{2} $
	71	$ 1 + 266. iT - 3.57e5T^{2} $
	73	$ 1 + 210. iT - 3.89e5T^{2} $
	79	$ 1 + 821. iT - 4.93e5T^{2} $
	83	$ 1 - 863.T + 5.71e5T^{2} $
	89	$ 1 - 1.02e3iT - 7.04e5T^{2} $
	97	$ 1 - 528. 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

−9.195997782379424109852793491192, −8.744717047329672469461510582163, −7.958555731000803305147164681765, −7.12211868710255329193606466638, −6.01151902973978633008649808150, −5.24717002297841792341361341731, −4.17673331272358168655905697713, −3.45311825485334332644187563364, −2.19049509184102906713243502453, −1.03660398372046778399737186206, 0.18280009148751236208563498377, 2.11308752855695730690211164389, 2.81842129631204712012086771740, 3.64508801910859527175832502974, 4.59089303639483670682073705077, 6.12357169579917501073450144026, 6.70371154648424020031044383690, 7.12951523720274126568889352826, 8.409376749943460116637393945410, 9.046815187933650635489881658074

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8195174541\)
\(L(\frac12)\)	\(\approx\)	\(0.8195174541\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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