LMFDB - L-function 2-1344-7.6-c2-0-23

Dirichlet series

L(s) = 1

− 1.73i·3^-s + 6.92i·5^-s + 7·7^-s − 2.99·9^-s + 18·11^-s + 20.7i·13^-s + 11.9·15^-s + 13.8i·17^-s + 20.7i·19^-s − 12.1i·21^-s − 18·23^-s − 22.9·25^-s + 5.19i·27^-s − 18·29^-s − 41.5i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.38i·5^-s + 7^-s − 0.333·9^-s + 1.63·11^-s + 1.59i·13^-s + 0.799·15^-s + 0.815i·17^-s + 1.09i·19^-s − 0.577i·21^-s − 0.782·23^-s − 0.919·25^-s + 0.192i·27^-s − 0.620·29^-s − 1.34i·31^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$-i$
Analytic conductor:	$36.6213$
Root analytic conductor:	$6.05155$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (769, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :1),\ -i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.141030686$
$L(\frac12)$	$\approx$	$2.141030686$
$L(2)$		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 + 1.73iT $
	7	$ 1 - 7T $
good	5	$ 1 - 6.92iT - 25T^{2} $
	11	$ 1 - 18T + 121T^{2} $
	13	$ 1 - 20.7iT - 169T^{2} $
	17	$ 1 - 13.8iT - 289T^{2} $
	19	$ 1 - 20.7iT - 361T^{2} $
	23	$ 1 + 18T + 529T^{2} $
	29	$ 1 + 18T + 841T^{2} $
	31	$ 1 + 41.5iT - 961T^{2} $
	37	$ 1 + 10T + 1.36e3T^{2} $
	41	$ 1 + 55.4iT - 1.68e3T^{2} $
	43	$ 1 + 38T + 1.84e3T^{2} $
	47	$ 1 - 27.7iT - 2.20e3T^{2} $
	53	$ 1 + 18T + 2.80e3T^{2} $
	59	$ 1 - 6.92iT - 3.48e3T^{2} $
	61	$ 1 - 20.7iT - 3.72e3T^{2} $
	67	$ 1 - 26T + 4.48e3T^{2} $
	71	$ 1 + 18T + 5.04e3T^{2} $
	73	$ 1 - 41.5iT - 5.32e3T^{2} $
	79	$ 1 + 2T + 6.24e3T^{2} $
	83	$ 1 + 34.6iT - 6.88e3T^{2} $
	89	$ 1 - 7.92e3T^{2} $
	97	$ 1 - 166. iT - 9.40e3T^{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

−9.602605104720060766718319243010, −8.755843016299506426781056845066, −7.86909167921907263944406842532, −7.08768311876702985391768761487, −6.44890969133572737378472561029, −5.81257097993710333034875811247, −4.19880774110073855862499470868, −3.70472489668587406584755416517, −2.10870947309227642330016853983, −1.58423021984063384616590059017, 0.62922715952750465348233519727, 1.59242228671090349878653680854, 3.18476358289176638389719905918, 4.28927662196642344201082297164, 4.96269063600235531505415596553, 5.52215884897579178714746600947, 6.74402876735082216259949767497, 7.891083066568777430352574356538, 8.546447340265239219768093611550, 9.130134389480657130215946112903

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(2.141030686\)
\(L(\frac12)\)	\(\approx\)	\(2.141030686\)
\(L(2)\)		not available
\(L(1)\)		not available

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