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Properties

Label 2-1280-40.13-c0-0-1
Degree $2$
Conductor $1280$
Sign $0.973 - 0.229i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + i·9-s + (1 − i)13-s + (−1 + i)17-s + 25-s + (−1 − i)37-s + i·45-s + i·49-s + (1 − i)53-s + (1 − i)65-s + (−1 − i)73-s − 81-s + (−1 + i)85-s + (1 − i)97-s + 2i·101-s + ⋯
L(s)  = 1  + 5-s + i·9-s + (1 − i)13-s + (−1 + i)17-s + 25-s + (−1 − i)37-s + i·45-s + i·49-s + (1 − i)53-s + (1 − i)65-s + (−1 − i)73-s − 81-s + (−1 + i)85-s + (1 − i)97-s + 2i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280444487\)
\(L(\frac12)\) \(\approx\) \(1.280444487\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−10.07403066264750029921991646453, −8.947732681856564842015412916729, −8.438466821913513519585770745763, −7.49173514819837414169334955315, −6.42925465942160716011031093663, −5.74474983610210005102651214319, −4.97789828717673226204301224537, −3.80848993559041850282377922754, −2.56382699017365319480480876623, −1.59728579618910801388038849945, 1.37436779908331870438816338532, 2.57835656525677766203580935029, 3.74021845670869840604791606181, 4.75463737771097494376872630462, 5.79187168655000644734836805734, 6.58780435151141343806501754820, 7.04457857340801992451861750460, 8.590450849705363797826252254669, 9.017766454665096541390124675676, 9.709306267011088429070222059352

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