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Properties

Label 2-1280-40.19-c2-0-56
Degree $2$
Conductor $1280$
Sign $0.551 - 0.834i$
Analytic cond. $34.8774$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·3-s + (4.89 − i)5-s + 8.48·7-s + 0.999·9-s + 13.8·11-s − 9.79·13-s + (2.82 + 13.8i)15-s + 19.5i·17-s + 13.8·19-s + 24i·21-s − 25.4·23-s + (22.9 − 9.79i)25-s + 28.2i·27-s + 22i·29-s − 55.4i·31-s + ⋯
L(s)  = 1  + 0.942i·3-s + (0.979 − 0.200i)5-s + 1.21·7-s + 0.111·9-s + 1.25·11-s − 0.753·13-s + (0.188 + 0.923i)15-s + 1.15i·17-s + 0.729·19-s + 1.14i·21-s − 1.10·23-s + (0.919 − 0.391i)25-s + 1.04i·27-s + 0.758i·29-s − 1.78i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.551 - 0.834i$
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: \((2,\ 1280,\ (\ :1),\ 0.551 - 0.834i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.145876160\)
\(L(\frac12)\) \(\approx\) \(3.145876160\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-4.89 + i)T \)
good3 \( 1 - 2.82iT - 9T^{2} \)
7 \( 1 - 8.48T + 49T^{2} \)
11 \( 1 - 13.8T + 121T^{2} \)
13 \( 1 + 9.79T + 169T^{2} \)
17 \( 1 - 19.5iT - 289T^{2} \)
19 \( 1 - 13.8T + 361T^{2} \)
23 \( 1 + 25.4T + 529T^{2} \)
29 \( 1 - 22iT - 841T^{2} \)
31 \( 1 + 55.4iT - 961T^{2} \)
37 \( 1 - 48.9T + 1.36e3T^{2} \)
41 \( 1 + 22T + 1.68e3T^{2} \)
43 \( 1 + 59.3iT - 1.84e3T^{2} \)
47 \( 1 + 8.48T + 2.20e3T^{2} \)
53 \( 1 - 29.3T + 2.80e3T^{2} \)
59 \( 1 - 13.8T + 3.48e3T^{2} \)
61 \( 1 + 46iT - 3.72e3T^{2} \)
67 \( 1 - 59.3iT - 4.48e3T^{2} \)
71 \( 1 + 27.7iT - 5.04e3T^{2} \)
73 \( 1 - 78.3iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 76.3iT - 6.88e3T^{2} \)
89 \( 1 + 146T + 7.92e3T^{2} \)
97 \( 1 - 58.7iT - 9.40e3T^{2} \)
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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

−9.732226814104469913738415830607, −8.981888266652868535972932466695, −8.157699629664109905464332023163, −7.15595952474542089820711615690, −6.07010218936440260426757388001, −5.30836330888118457617310722526, −4.44571365453307665004385943114, −3.80315076817805001793510883276, −2.17426145890645585376385509350, −1.31518507967633625688422412295, 1.09334268446599956154775636220, 1.77626216620086984927337290820, 2.77971990274096894626213068468, 4.35791111000074101415964790600, 5.17092425244468525499847814463, 6.18571430528008217238575846916, 6.92790664554939513170874575757, 7.58403022847369281922070461872, 8.426889061420562905958212214413, 9.499933032268202167700698776804

Graph of the $Z$-function along the critical line