LMFDB - L-function 2-42e2-21.20-c3-0-29

Dirichlet series

L(s) = 1

+ 20.2·5^-s − 17.8i·11^-s − 33.1i·13^-s + 45.8·17^-s + 40.4i·19^-s + 80.5i·23^-s + 285.·25^-s + 233. i·29^-s − 225. i·31^-s − 270.·37^-s − 154.·41^-s + 367.·43^-s + 527.·47^-s + 91.0i·53^-s − 361. i·55^-s + ⋯

L(s) = 1

+ 1.81·5^-s − 0.489i·11^-s − 0.707i·13^-s + 0.653·17^-s + 0.488i·19^-s + 0.730i·23^-s + 2.28·25^-s + 1.49i·29^-s − 1.30i·31^-s − 1.20·37^-s − 0.588·41^-s + 1.30·43^-s + 1.63·47^-s + 0.236i·53^-s − 0.885i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1764$ = $2^{2} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.970 + 0.239i$
Analytic conductor:	$104.079$
Root analytic conductor:	$10.2019$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1764} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1764,\ (\ :3/2),\ 0.970 + 0.239i)$

Particular Values

$L(2)$	$\approx$	$3.506767781$
$L(\frac12)$	$\approx$	$3.506767781$
$L(\frac{5}{2})$		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 $
	7	$ 1 $
good	5	$ 1 - 20.2T + 125T^{2} $
	11	$ 1 + 17.8iT - 1.33e3T^{2} $
	13	$ 1 + 33.1iT - 2.19e3T^{2} $
	17	$ 1 - 45.8T + 4.91e3T^{2} $
	19	$ 1 - 40.4iT - 6.85e3T^{2} $
	23	$ 1 - 80.5iT - 1.21e4T^{2} $
	29	$ 1 - 233. iT - 2.43e4T^{2} $
	31	$ 1 + 225. iT - 2.97e4T^{2} $
	37	$ 1 + 270.T + 5.06e4T^{2} $
	41	$ 1 + 154.T + 6.89e4T^{2} $
	43	$ 1 - 367.T + 7.95e4T^{2} $
	47	$ 1 - 527.T + 1.03e5T^{2} $
	53	$ 1 - 91.0iT - 1.48e5T^{2} $
	59	$ 1 - 625.T + 2.05e5T^{2} $
	61	$ 1 - 91.0iT - 2.26e5T^{2} $
	67	$ 1 - 863.T + 3.00e5T^{2} $
	71	$ 1 + 303. iT - 3.57e5T^{2} $
	73	$ 1 + 1.15e3iT - 3.89e5T^{2} $
	79	$ 1 + 6.96T + 4.93e5T^{2} $
	83	$ 1 - 815.T + 5.71e5T^{2} $
	89	$ 1 + 310.T + 7.04e5T^{2} $
	97	$ 1 - 1.83e3iT - 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

−9.120647192615486873836263938314, −8.206105932811985470015452276268, −7.25120349598334471014193880566, −6.33418571238727342578189155780, −5.52678543584489335291864543782, −5.29619355569583217315475313464, −3.74394413258088565150067338607, −2.78202759538312579671282110490, −1.82622261130347503317429335828, −0.879635653947831141743409782122, 0.969201039364374847722724409065, 2.02360962127613823371949094189, 2.64521869444521504394315074016, 4.04337566461313070491130708407, 5.09486346033856048234944839696, 5.68480241065800144826746953752, 6.60395285161162168530382871699, 7.09690200588232621859445568247, 8.401695612967482664119814754266, 9.086221470324012107567826797633

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.506767781\)
\(L(\frac12)\)	\(\approx\)	\(3.506767781\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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