LMFDB - L-function 2-21e2-21.20-c3-0-6

Dirichlet series

L(s) = 1

− 0.748i·2^-s + 7.43·4^-s − 10.8·5^-s − 11.5i·8^-s + 8.13i·10^-s + 51.8i·11^-s − 32.1i·13^-s + 50.8·16^-s − 81.4·17^-s + 0.0485i·19^-s − 80.7·20^-s + 38.8·22^-s + 89.2i·23^-s − 7.16·25^-s − 24.1·26^-s + ⋯

L(s) = 1

− 0.264i·2^-s + 0.929·4^-s − 0.970·5^-s − 0.511i·8^-s + 0.257i·10^-s + 1.42i·11^-s − 0.686i·13^-s + 0.794·16^-s − 1.16·17^-s + 0.000586i·19^-s − 0.902·20^-s + 0.376·22^-s + 0.809i·23^-s − 0.0572·25^-s − 0.181·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$441$ = $3^{2} \cdot 7^{2}$
Sign:	$-0.0980 - 0.995i$
Analytic conductor:	$26.0198$
Root analytic conductor:	$5.10096$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{441} (440, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 441,\ (\ :3/2),\ -0.0980 - 0.995i)$

Particular Values

$L(2)$	$\approx$	$1.150195436$
$L(\frac12)$	$\approx$	$1.150195436$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 $
good	2	$ 1 + 0.748iT - 8T^{2} $
	5	$ 1 + 10.8T + 125T^{2} $
	11	$ 1 - 51.8iT - 1.33e3T^{2} $
	13	$ 1 + 32.1iT - 2.19e3T^{2} $
	17	$ 1 + 81.4T + 4.91e3T^{2} $
	19	$ 1 - 0.0485iT - 6.85e3T^{2} $
	23	$ 1 - 89.2iT - 1.21e4T^{2} $
	29	$ 1 - 175. iT - 2.43e4T^{2} $
	31	$ 1 - 215. iT - 2.97e4T^{2} $
	37	$ 1 - 64.5T + 5.06e4T^{2} $
	41	$ 1 - 411.T + 6.89e4T^{2} $
	43	$ 1 + 234.T + 7.95e4T^{2} $
	47	$ 1 + 632.T + 1.03e5T^{2} $
	53	$ 1 - 265. iT - 1.48e5T^{2} $
	59	$ 1 + 351.T + 2.05e5T^{2} $
	61	$ 1 - 778. iT - 2.26e5T^{2} $
	67	$ 1 + 196.T + 3.00e5T^{2} $
	71	$ 1 + 142. iT - 3.57e5T^{2} $
	73	$ 1 - 780. iT - 3.89e5T^{2} $
	79	$ 1 - 1.28e3T + 4.93e5T^{2} $
	83	$ 1 + 235.T + 5.71e5T^{2} $
	89	$ 1 - 670.T + 7.04e5T^{2} $
	97	$ 1 - 655. 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

−11.00516244825434336102971113381, −10.27856133004559348697213738589, −9.228630600559675000029907653307, −7.962771831784966975254957399352, −7.29346624544454290230892261992, −6.51595613922824756576015209240, −5.07204580440954550128256951102, −3.94213128353435459353715128895, −2.81143604159024935485161784613, −1.51458313742876646140496860714, 0.35192754438797187297000799858, 2.18880142974107168518900333316, 3.43356332181931404657529683054, 4.53786610134726860750502131022, 6.04607969197062423154463262393, 6.62103767952697417623107120459, 7.82534614607397762168378385860, 8.313939956967395361744715359272, 9.484761295938560174821789942793, 10.88802387846369505356762073412

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.150195436\)
\(L(\frac12)\)	\(\approx\)	\(1.150195436\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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