LMFDB - L-function 2-1344-56.27-c3-0-31

Dirichlet series

L(s) = 1

− 3i·3^-s − 15.6·5^-s + (10.8 − 15.0i)7^-s − 9·9^-s − 41.3·11^-s − 8.02·13^-s + 47.0i·15^-s + 14.2i·17^-s + 10.2i·19^-s + (−45.0 − 32.5i)21^-s + 203. i·23^-s + 121.·25^-s + 27i·27^-s − 82.6i·29^-s − 156.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.40·5^-s + (0.585 − 0.810i)7^-s − 0.333·9^-s − 1.13·11^-s − 0.171·13^-s + 0.810i·15^-s + 0.202i·17^-s + 0.123i·19^-s + (−0.468 − 0.337i)21^-s + 1.84i·23^-s + 0.969·25^-s + 0.192i·27^-s − 0.529i·29^-s − 0.905·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.934 + 0.355i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.934 + 0.355i)$

Particular Values

$L(2)$	$\approx$	$0.9447960393$
$L(\frac12)$	$\approx$	$0.9447960393$
$L(\frac{5}{2})$		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 + (-10.8 + 15.0i)T $
good	5	$ 1 + 15.6T + 125T^{2} $
	11	$ 1 + 41.3T + 1.33e3T^{2} $
	13	$ 1 + 8.02T + 2.19e3T^{2} $
	17	$ 1 - 14.2iT - 4.91e3T^{2} $
	19	$ 1 - 10.2iT - 6.85e3T^{2} $
	23	$ 1 - 203. iT - 1.21e4T^{2} $
	29	$ 1 + 82.6iT - 2.43e4T^{2} $
	31	$ 1 + 156.T + 2.97e4T^{2} $
	37	$ 1 + 106. iT - 5.06e4T^{2} $
	41	$ 1 - 315. iT - 6.89e4T^{2} $
	43	$ 1 + 154.T + 7.95e4T^{2} $
	47	$ 1 - 474.T + 1.03e5T^{2} $
	53	$ 1 + 266. iT - 1.48e5T^{2} $
	59	$ 1 - 425. iT - 2.05e5T^{2} $
	61	$ 1 - 8.30T + 2.26e5T^{2} $
	67	$ 1 + 820.T + 3.00e5T^{2} $
	71	$ 1 + 109. iT - 3.57e5T^{2} $
	73	$ 1 + 53.8iT - 3.89e5T^{2} $
	79	$ 1 + 1.17e3iT - 4.93e5T^{2} $
	83	$ 1 + 507. iT - 5.71e5T^{2} $
	89	$ 1 - 1.06e3iT - 7.04e5T^{2} $
	97	$ 1 + 573. 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

−8.992266955210312673690539746162, −7.991524769588034878228507323429, −7.62417596859111501670004448977, −7.18671994484372514846668920412, −5.84460300079165421202283779463, −4.90159083950387057664137915731, −3.98414524561833224729436922120, −3.15328238985842891946339446088, −1.76751411036079101309442441614, −0.53518559102915963754524682754, 0.41892334411207087322547386351, 2.29051675480980030966886202152, 3.18031780707637030384027947228, 4.28154670502317796018983664630, 4.92015971155414007927952970103, 5.74097602123571420633655200581, 7.03988766992872363850467961057, 7.81892147641156643178692469315, 8.497631461080713311371988310819, 9.035137180625889994992737651498

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9447960393\)
\(L(\frac12)\)	\(\approx\)	\(0.9447960393\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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