LMFDB - L-function 2-1344-28.27-c3-0-17

Dirichlet series

L(s) = 1

− 3·3^-s − 12.4i·5^-s + (−8.98 − 16.1i)7^-s + 9·9^-s + 66.0i·11^-s − 82.6i·13^-s + 37.4i·15^-s + 30.0i·17^-s + 4.86·19^-s + (26.9 + 48.5i)21^-s + 113. i·23^-s − 31.0·25^-s − 27·27^-s − 233.·29^-s + 100.·31^-s + ⋯

L(s) = 1

− 0.577·3^-s − 1.11i·5^-s + (−0.485 − 0.874i)7^-s + 0.333·9^-s + 1.81i·11^-s − 1.76i·13^-s + 0.645i·15^-s + 0.428i·17^-s + 0.0587·19^-s + (0.280 + 0.504i)21^-s + 1.02i·23^-s − 0.248·25^-s − 0.192·27^-s − 1.49·29^-s + 0.583·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.874 - 0.485i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.874 - 0.485i)$

Particular Values

$L(2)$	$\approx$	$0.9698425450$
$L(\frac12)$	$\approx$	$0.9698425450$
$L(\frac{5}{2})$		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 + (8.98 + 16.1i)T $
good	5	$ 1 + 12.4iT - 125T^{2} $
	11	$ 1 - 66.0iT - 1.33e3T^{2} $
	13	$ 1 + 82.6iT - 2.19e3T^{2} $
	17	$ 1 - 30.0iT - 4.91e3T^{2} $
	19	$ 1 - 4.86T + 6.85e3T^{2} $
	23	$ 1 - 113. iT - 1.21e4T^{2} $
	29	$ 1 + 233.T + 2.43e4T^{2} $
	31	$ 1 - 100.T + 2.97e4T^{2} $
	37	$ 1 + 157.T + 5.06e4T^{2} $
	41	$ 1 - 433. iT - 6.89e4T^{2} $
	43	$ 1 + 217. iT - 7.95e4T^{2} $
	47	$ 1 + 328.T + 1.03e5T^{2} $
	53	$ 1 - 117.T + 1.48e5T^{2} $
	59	$ 1 - 384.T + 2.05e5T^{2} $
	61	$ 1 - 87.8iT - 2.26e5T^{2} $
	67	$ 1 + 305. iT - 3.00e5T^{2} $
	71	$ 1 - 1.07e3iT - 3.57e5T^{2} $
	73	$ 1 + 239. iT - 3.89e5T^{2} $
	79	$ 1 - 6.66iT - 4.93e5T^{2} $
	83	$ 1 + 443.T + 5.71e5T^{2} $
	89	$ 1 + 113. iT - 7.04e5T^{2} $
	97	$ 1 - 356. 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

−9.665149072816384643798171077990, −8.408358893095703500967931759758, −7.58414151681120734577623878826, −6.99743672420403088475584730495, −5.81265249052097171344718884581, −5.08163475665657647388085624519, −4.36135690657392138302146036656, −3.36059360126143306785074638395, −1.73510428748820005028839444177, −0.77921477037356068364799362414, 0.32890842124729047043928039166, 2.01253798741764469181282687498, 3.01393136298718888302520321045, 3.85581041995913735707483777890, 5.14897232717313890608661441906, 6.10545779319083616609087739394, 6.50832210424718053286385006184, 7.28556151320060422272317742531, 8.581199580722237908084453795522, 9.105003442225572080880065801755

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9698425450\)
\(L(\frac12)\)	\(\approx\)	\(0.9698425450\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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