LMFDB - L-function 2-21e2-21.20-c3-0-9

Dirichlet series

L(s) = 1

− 4.86i·2^-s − 15.6·4^-s + 12.7·5^-s + 37.3i·8^-s − 62.1i·10^-s + 54.1i·11^-s − 8.85i·13^-s + 56.2·16^-s − 68.9·17^-s + 163. i·19^-s − 200.·20^-s + 263.·22^-s + 93.9i·23^-s + 37.9·25^-s − 43.0·26^-s + ⋯

L(s) = 1

− 1.72i·2^-s − 1.95·4^-s + 1.14·5^-s + 1.64i·8^-s − 1.96i·10^-s + 1.48i·11^-s − 0.188i·13^-s + 0.878·16^-s − 0.983·17^-s + 1.97i·19^-s − 2.23·20^-s + 2.55·22^-s + 0.851i·23^-s + 0.303·25^-s − 0.324·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$441$ = $3^{2} \cdot 7^{2}$
Sign:	$0.970 + 0.239i$
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:	$(2,\ 441,\ (\ :3/2),\ 0.970 + 0.239i)$

Particular Values

$L(2)$	$\approx$	$1.410882887$
$L(\frac12)$	$\approx$	$1.410882887$
$L(\frac{5}{2})$		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 + 4.86iT - 8T^{2} $
	5	$ 1 - 12.7T + 125T^{2} $
	11	$ 1 - 54.1iT - 1.33e3T^{2} $
	13	$ 1 + 8.85iT - 2.19e3T^{2} $
	17	$ 1 + 68.9T + 4.91e3T^{2} $
	19	$ 1 - 163. iT - 6.85e3T^{2} $
	23	$ 1 - 93.9iT - 1.21e4T^{2} $
	29	$ 1 - 119. iT - 2.43e4T^{2} $
	31	$ 1 - 98.8iT - 2.97e4T^{2} $
	37	$ 1 - 94.1T + 5.06e4T^{2} $
	41	$ 1 + 259.T + 6.89e4T^{2} $
	43	$ 1 - 5.01T + 7.95e4T^{2} $
	47	$ 1 - 57.3T + 1.03e5T^{2} $
	53	$ 1 + 470. iT - 1.48e5T^{2} $
	59	$ 1 + 225.T + 2.05e5T^{2} $
	61	$ 1 + 427. iT - 2.26e5T^{2} $
	67	$ 1 - 163.T + 3.00e5T^{2} $
	71	$ 1 - 79.8iT - 3.57e5T^{2} $
	73	$ 1 - 769. iT - 3.89e5T^{2} $
	79	$ 1 - 534.T + 4.93e5T^{2} $
	83	$ 1 + 438.T + 5.71e5T^{2} $
	89	$ 1 - 25.6T + 7.04e5T^{2} $
	97	$ 1 + 1.38e3iT - 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

−10.52481435102930202137016601533, −9.904598294337053780824817140436, −9.468781243537649288597415035291, −8.339812268258759801740043721572, −6.91231467783596344504543890415, −5.59066212245572597537386551072, −4.57252935369952333596068487636, −3.44320946790626596917797226243, −2.06996749742623482464346176783, −1.57907564301414753598083149391, 0.44598510318722759026548179506, 2.56048617392880886706100865533, 4.37410610998421755377302318891, 5.35049383616386508559796162554, 6.23387255209409937449683712671, 6.69150209524961459786784402938, 7.918512436960003789511852408686, 8.935365689241395186151420484293, 9.232629546096772291924012418964, 10.57226748267777185777815288385

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.410882887\)
\(L(\frac12)\)	\(\approx\)	\(1.410882887\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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