LMFDB - L-function 2-1280-20.19-c2-0-4

Dirichlet series

L(s) = 1

− 3.74·3^-s + (−4.58 − 2i)5^-s + 2.44·7^-s + 5·9^-s + 14.6i·11^-s + 8i·13^-s + (17.1 + 7.48i)15^-s − 18.3i·17^-s − 24.4i·19^-s − 9.16·21^-s + 26.9·23^-s + (17 + 18.3i)25^-s + 14.9·27^-s − 18.3·29^-s − 44.8i·31^-s + ⋯

L(s) = 1

− 1.24·3^-s + (−0.916 − 0.400i)5^-s + 0.349·7^-s + 0.555·9^-s + 1.33i·11^-s + 0.615i·13^-s + (1.14 + 0.498i)15^-s − 1.07i·17^-s − 1.28i·19^-s − 0.436·21^-s + 1.17·23^-s + (0.680 + 0.733i)25^-s + 0.554·27^-s − 0.632·29^-s − 1.44i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1280$ = $2^{8} \cdot 5$
Sign:	$-0.916 - 0.399i$
Analytic conductor:	$34.8774$
Root analytic conductor:	$5.90571$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1280} (1279, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1280,\ (\ :1),\ -0.916 - 0.399i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.1277073823$
$L(\frac12)$	$\approx$	$0.1277073823$
$L(2)$		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 + (4.58 + 2i)T $
good	3	$ 1 + 3.74T + 9T^{2} $
	7	$ 1 - 2.44T + 49T^{2} $
	11	$ 1 - 14.6iT - 121T^{2} $
	13	$ 1 - 8iT - 169T^{2} $
	17	$ 1 + 18.3iT - 289T^{2} $
	19	$ 1 + 24.4iT - 361T^{2} $
	23	$ 1 - 26.9T + 529T^{2} $
	29	$ 1 + 18.3T + 841T^{2} $
	31	$ 1 + 44.8iT - 961T^{2} $
	37	$ 1 - 44iT - 1.36e3T^{2} $
	41	$ 1 - 20T + 1.68e3T^{2} $
	43	$ 1 + 11.2T + 1.84e3T^{2} $
	47	$ 1 - 41.6T + 2.20e3T^{2} $
	53	$ 1 + 32iT - 2.80e3T^{2} $
	59	$ 1 + 34.2iT - 3.48e3T^{2} $
	61	$ 1 + 64.1T + 3.72e3T^{2} $
	67	$ 1 - 78.5T + 4.48e3T^{2} $
	71	$ 1 - 134. iT - 5.04e3T^{2} $
	73	$ 1 - 91.6iT - 5.32e3T^{2} $
	79	$ 1 - 44.8iT - 6.24e3T^{2} $
	83	$ 1 + 11.2T + 6.88e3T^{2} $
	89	$ 1 + 10T + 7.92e3T^{2} $
	97	$ 1 + 54.9iT - 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

−9.763555438387296559229615025966, −9.169161012935814601453191311944, −8.099492186820993619123722637458, −7.08136084001261899308110326986, −6.81050225216258285229725806968, −5.36809261695253128236792191697, −4.81621920512564680553634185728, −4.19933580898938483919126020404, −2.62956715288129550871863962349, −1.07424333495899668601533628962, 0.05820069957192854437417752979, 1.21913692517230316690008216521, 3.09392736589719207303767429748, 3.88515399342862627558454837059, 5.05045346773489348928812898320, 5.81230838777472403067862387988, 6.43911909096744783572379368964, 7.50850419864041328468417852912, 8.213097312236498766846614377277, 8.973930655560697874577573246313

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1277073823\)
\(L(\frac12)\)	\(\approx\)	\(0.1277073823\)
\(L(2)\)		not available
\(L(1)\)		not available

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