LMFDB - L-function 2-1280-40.19-c2-0-49

Dirichlet series

L(s) = 1

− 4.47i·3^-s + 5i·5^-s + 13.4·7^-s − 11.0·9^-s + 22.3·15^-s − 60.0i·21^-s + 13.4·23^-s − 25·25^-s + 8.94i·27^-s + 22i·29^-s + 67.0i·35^-s + 62·41^-s + 40.2i·43^-s − 55.0i·45^-s + 93.9·47^-s + ⋯

L(s) = 1

− 1.49i·3^-s + i·5^-s + 1.91·7^-s − 1.22·9^-s + 1.49·15^-s − 2.85i·21^-s + 0.583·23^-s − 25^-s + 0.331i·27^-s + 0.758i·29^-s + 1.91i·35^-s + 1.51·41^-s + 0.936i·43^-s − 1.22i·45^-s + 1.99·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1280$ = $2^{8} \cdot 5$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$34.8774$
Root analytic conductor:	$5.90571$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1280} (639, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1280,\ (\ :1),\ 0.707 + 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.564063229$
$L(\frac12)$	$\approx$	$2.564063229$
$L(2)$		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 $
	5	$ 1 - 5iT $
good	3	$ 1 + 4.47iT - 9T^{2} $
	7	$ 1 - 13.4T + 49T^{2} $
	11	$ 1 + 121T^{2} $
	13	$ 1 + 169T^{2} $
	17	$ 1 - 289T^{2} $
	19	$ 1 + 361T^{2} $
	23	$ 1 - 13.4T + 529T^{2} $
	29	$ 1 - 22iT - 841T^{2} $
	31	$ 1 - 961T^{2} $
	37	$ 1 + 1.36e3T^{2} $
	41	$ 1 - 62T + 1.68e3T^{2} $
	43	$ 1 - 40.2iT - 1.84e3T^{2} $
	47	$ 1 - 93.9T + 2.20e3T^{2} $
	53	$ 1 + 2.80e3T^{2} $
	59	$ 1 + 3.48e3T^{2} $
	61	$ 1 + 58iT - 3.72e3T^{2} $
	67	$ 1 - 67.0iT - 4.48e3T^{2} $
	71	$ 1 - 5.04e3T^{2} $
	73	$ 1 - 5.32e3T^{2} $
	79	$ 1 - 6.24e3T^{2} $
	83	$ 1 + 147. iT - 6.88e3T^{2} $
	89	$ 1 - 142T + 7.92e3T^{2} $
	97	$ 1 - 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

−9.137331142733776164337563449837, −8.231706643785272328646762454495, −7.59026137976584722414139290692, −7.17267196034562518195384566413, −6.21441538251241973290134243868, −5.35089378344739684649375227426, −4.22809411274490126313145621120, −2.75406427973830161945788341699, −1.93722412508759310270914217975, −1.01991485144524085662118700402, 1.00054258564060164681877071369, 2.31182674310274815769606450899, 3.95722945850504871039277967285, 4.44050091372220930313406763190, 5.16774888551262673665676737989, 5.71871526262673651444951719747, 7.43671265336181436044783384379, 8.193654329903563627108170682000, 8.906156288670725302775547260727, 9.417597999092150494735456995645

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.564063229\)
\(L(\frac12)\)	\(\approx\)	\(2.564063229\)
\(L(2)\)		not available
\(L(1)\)		not available

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