LMFDB - L-function 2-21e2-7.6-c6-0-19

Dirichlet series

L(s) = 1

− 12.4·2^-s + 91.6·4^-s + 202. i·5^-s − 345.·8^-s − 2.52e3i·10^-s − 875.·11^-s − 275. i·13^-s − 1.56e3·16^-s − 4.38e3i·17^-s − 1.35e4i·19^-s + 1.85e4i·20^-s + 1.09e4·22^-s + 1.27e4·23^-s − 2.53e4·25^-s + 3.43e3i·26^-s + ⋯

L(s) = 1

− 1.55·2^-s + 1.43·4^-s + 1.61i·5^-s − 0.674·8^-s − 2.52i·10^-s − 0.657·11^-s − 0.125i·13^-s − 0.380·16^-s − 0.892i·17^-s − 1.96i·19^-s + 2.32i·20^-s + 1.02·22^-s + 1.04·23^-s − 1.62·25^-s + 0.195i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$441$ = $3^{2} \cdot 7^{2}$
Sign:	$-0.755 - 0.654i$
Analytic conductor:	$101.453$
Root analytic conductor:	$10.0724$
Motivic weight:	$6$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{441} (244, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 441,\ (\ :3),\ -0.755 - 0.654i)$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.5181701835$
$L(\frac12)$	$\approx$	$0.5181701835$
$L(4)$		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 + 12.4T + 64T^{2} $
	5	$ 1 - 202. iT - 1.56e4T^{2} $
	11	$ 1 + 875.T + 1.77e6T^{2} $
	13	$ 1 + 275. iT - 4.82e6T^{2} $
	17	$ 1 + 4.38e3iT - 2.41e7T^{2} $
	19	$ 1 + 1.35e4iT - 4.70e7T^{2} $
	23	$ 1 - 1.27e4T + 1.48e8T^{2} $
	29	$ 1 + 6.26e3T + 5.94e8T^{2} $
	31	$ 1 - 2.04e4iT - 8.87e8T^{2} $
	37	$ 1 - 1.57e4T + 2.56e9T^{2} $
	41	$ 1 - 6.99e4iT - 4.75e9T^{2} $
	43	$ 1 - 1.13e5T + 6.32e9T^{2} $
	47	$ 1 - 4.63e4iT - 1.07e10T^{2} $
	53	$ 1 + 1.28e5T + 2.21e10T^{2} $
	59	$ 1 - 2.80e5iT - 4.21e10T^{2} $
	61	$ 1 - 9.78e4iT - 5.15e10T^{2} $
	67	$ 1 - 1.74e5T + 9.04e10T^{2} $
	71	$ 1 + 3.45e5T + 1.28e11T^{2} $
	73	$ 1 - 1.20e5iT - 1.51e11T^{2} $
	79	$ 1 - 7.63e5T + 2.43e11T^{2} $
	83	$ 1 + 8.59e5iT - 3.26e11T^{2} $
	89	$ 1 - 1.79e5iT - 4.96e11T^{2} $
	97	$ 1 - 3.40e5iT - 8.32e11T^{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

−10.49396479677946544861463812894, −9.551566209193554667604705327610, −8.854192776177552563876649069374, −7.56177308051486554909797281743, −7.20066177162531329202792913777, −6.35131573602611267369535963694, −4.81994913237567967883429337018, −2.95471828625692777197821580448, −2.48065595633735548272040943308, −0.861038837306475752056865618136, 0.26219142950760228094772109147, 1.21521547605773007440159677235, 2.04222945828047954588374468268, 3.91770868662727004381939029978, 5.11490170727240710707606137800, 6.12794285977038224283560023399, 7.60889233468311135503917648861, 8.140309970256292557747412599394, 8.865050743862281125524931617769, 9.590110151425438799717597364464

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.5181701835\)
\(L(\frac12)\)	\(\approx\)	\(0.5181701835\)
\(L(4)\)		not available
\(L(1)\)		not available

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