LMFDB - L-function 2-1344-8.5-c3-0-23

Dirichlet series

L(s) = 1

− 3i·3^-s − 13.8i·5^-s − 7·7^-s − 9·9^-s + 9.03i·11^-s + 50.6i·13^-s − 41.4·15^-s + 1.30·17^-s − 14.9i·19^-s + 21i·21^-s + 62.4·23^-s − 65.4·25^-s + 27i·27^-s + 37.3i·29^-s + 32.6·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.23i·5^-s − 0.377·7^-s − 0.333·9^-s + 0.247i·11^-s + 1.08i·13^-s − 0.712·15^-s + 0.0185·17^-s − 0.179i·19^-s + 0.218i·21^-s + 0.566·23^-s − 0.523·25^-s + 0.192i·27^-s + 0.239i·29^-s + 0.189·31^-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

$L(2)$	$\approx$	$1.719619352$
$L(\frac12)$	$\approx$	$1.719619352$
$L(\frac{5}{2})$		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 + 13.8iT - 125T^{2} $
	11	$ 1 - 9.03iT - 1.33e3T^{2} $
	13	$ 1 - 50.6iT - 2.19e3T^{2} $
	17	$ 1 - 1.30T + 4.91e3T^{2} $
	19	$ 1 + 14.9iT - 6.85e3T^{2} $
	23	$ 1 - 62.4T + 1.21e4T^{2} $
	29	$ 1 - 37.3iT - 2.43e4T^{2} $
	31	$ 1 - 32.6T + 2.97e4T^{2} $
	37	$ 1 - 307. iT - 5.06e4T^{2} $
	41	$ 1 - 19.4T + 6.89e4T^{2} $
	43	$ 1 - 201. iT - 7.95e4T^{2} $
	47	$ 1 - 273.T + 1.03e5T^{2} $
	53	$ 1 - 29.5iT - 1.48e5T^{2} $
	59	$ 1 - 625. iT - 2.05e5T^{2} $
	61	$ 1 - 758. iT - 2.26e5T^{2} $
	67	$ 1 + 733. iT - 3.00e5T^{2} $
	71	$ 1 - 107.T + 3.57e5T^{2} $
	73	$ 1 - 342.T + 3.89e5T^{2} $
	79	$ 1 + 800.T + 4.93e5T^{2} $
	83	$ 1 + 1.09e3iT - 5.71e5T^{2} $
	89	$ 1 - 1.29e3T + 7.04e5T^{2} $
	97	$ 1 + 301.T + 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

−9.033183842621556675698039531372, −8.578061350998869174277426230766, −7.55809290297501410083586995300, −6.78412540110566758023661471538, −5.94040140859615407092714939510, −4.92277630583516072395153421191, −4.25338078335707345890186380447, −2.92459683096101323200951454391, −1.69990180780179958313299554640, −0.828833991506555483435103294368, 0.52525914718933606347808196353, 2.36631888959083585538433149702, 3.19290706400052909802432241535, 3.86921993547637105263116948066, 5.18053451529338409547860831523, 5.95425977483547503631934657436, 6.80428835442147887435953264623, 7.59632824141741111212932371626, 8.488951741218322546116840466166, 9.448099158499960544985016072842

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.719619352\)
\(L(\frac12)\)	\(\approx\)	\(1.719619352\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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