LMFDB - L-function 2-1344-56.27-c3-0-44

Dirichlet series

L(s) = 1

+ 3i·3^-s + 10.6·5^-s + (−18.0 − 4.29i)7^-s − 9·9^-s − 5.02·11^-s − 80.6·13^-s + 31.9i·15^-s − 31.3i·17^-s − 74.0i·19^-s + (12.8 − 54.0i)21^-s − 37.3i·23^-s − 11.9·25^-s − 27i·27^-s + 242. i·29^-s + 284.·31^-s + ⋯

L(s) = 1

+ 0.577i·3^-s + 0.951·5^-s + (−0.972 − 0.231i)7^-s − 0.333·9^-s − 0.137·11^-s − 1.72·13^-s + 0.549i·15^-s − 0.446i·17^-s − 0.894i·19^-s + (0.133 − 0.561i)21^-s − 0.338i·23^-s − 0.0952·25^-s − 0.192i·27^-s + 1.55i·29^-s + 1.65·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.851 - 0.523i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.851 - 0.523i)$

Particular Values

$L(2)$	$\approx$	$1.739542977$
$L(\frac12)$	$\approx$	$1.739542977$
$L(\frac{5}{2})$		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 + (18.0 + 4.29i)T $
good	5	$ 1 - 10.6T + 125T^{2} $
	11	$ 1 + 5.02T + 1.33e3T^{2} $
	13	$ 1 + 80.6T + 2.19e3T^{2} $
	17	$ 1 + 31.3iT - 4.91e3T^{2} $
	19	$ 1 + 74.0iT - 6.85e3T^{2} $
	23	$ 1 + 37.3iT - 1.21e4T^{2} $
	29	$ 1 - 242. iT - 2.43e4T^{2} $
	31	$ 1 - 284.T + 2.97e4T^{2} $
	37	$ 1 + 167. iT - 5.06e4T^{2} $
	41	$ 1 - 270. iT - 6.89e4T^{2} $
	43	$ 1 - 169.T + 7.95e4T^{2} $
	47	$ 1 - 639.T + 1.03e5T^{2} $
	53	$ 1 - 534. iT - 1.48e5T^{2} $
	59	$ 1 + 56.9iT - 2.05e5T^{2} $
	61	$ 1 - 217.T + 2.26e5T^{2} $
	67	$ 1 - 684.T + 3.00e5T^{2} $
	71	$ 1 - 966. iT - 3.57e5T^{2} $
	73	$ 1 + 325. iT - 3.89e5T^{2} $
	79	$ 1 + 856. iT - 4.93e5T^{2} $
	83	$ 1 - 459. iT - 5.71e5T^{2} $
	89	$ 1 + 1.07e3iT - 7.04e5T^{2} $
	97	$ 1 + 1.22e3iT - 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

−9.400772155437696117454008566204, −8.889661286156880596268145733806, −7.51703598817050675301946396466, −6.84712320405522210648544935115, −5.93548320504889367012836170656, −5.09007620917933443813678890513, −4.32081757122311447660696258771, −2.93770006619219803419855935923, −2.41783478259178314016931094204, −0.66859531354571479024465599497, 0.59841437853977594745229866064, 2.11207404582182727511040857521, 2.61644935512390839684688088961, 3.93465001566691378662722942642, 5.22261597684010742211323735519, 5.95973170066222163846981439777, 6.59294367145884927048535107979, 7.50040878157384439871868544575, 8.287267327322709271823016005448, 9.410371808488057823498838312493

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.739542977\)
\(L(\frac12)\)	\(\approx\)	\(1.739542977\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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