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Properties

Label 2-1280-5.2-c0-0-0
Degree $2$
Conductor $1280$
Sign $0.850 - 0.525i$
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 + 2i·29-s + (1 + i)37-s − i·45-s − i·49-s + (1 − i)53-s − 2·61-s + (−1 + i)65-s + (1 − i)73-s − 81-s + (−1 − i)85-s + ⋯
L(s)  = 1  − 5-s + i·9-s + (1 − i)13-s + (1 + i)17-s + 25-s + 2i·29-s + (1 + i)37-s − i·45-s − i·49-s + (1 − i)53-s − 2·61-s + (−1 + i)65-s + (1 − i)73-s − 81-s + (−1 − i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9419259409\)
\(L(\frac12)\) \(\approx\) \(0.9419259409\)
\(L(1)\) 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 \( 1 - 2iT - T^{2} \)
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 \( 1 + 2T + T^{2} \)
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

−10.23711400181185350561866138118, −8.886505133414054586252532406794, −8.146170534884785783734466792720, −7.77302220976282051780413503769, −6.73332600072160857764650719305, −5.65653986091227342457376095456, −4.87939390352804432372898037592, −3.75065756697698574979972435297, −3.03584198863477598145321032218, −1.37751409789178852066729876465, 0.983393780236680096951006918568, 2.77534599961601714499993196527, 3.84261966850104562744588414904, 4.36930217022453993242359497317, 5.74152591236722501722998041618, 6.51986457341453147936401563017, 7.42542116703009233902558842530, 8.092599551787920567021759210917, 9.118745913911783528720798821165, 9.541016757058697470036630218576

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