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Properties

Label 2-1280-16.13-c1-0-0
Degree $2$
Conductor $1280$
Sign $-0.991 - 0.130i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.341 + 0.341i)3-s + (−0.707 + 0.707i)5-s − 1.89i·7-s − 2.76i·9-s + (−3.24 + 3.24i)11-s + (2.93 + 2.93i)13-s − 0.482·15-s − 8.00·17-s + (−3.98 − 3.98i)19-s + (0.646 − 0.646i)21-s + 5.61i·23-s − 1.00i·25-s + (1.96 − 1.96i)27-s + (1.39 + 1.39i)29-s − 6.29·31-s + ⋯
L(s)  = 1  + (0.196 + 0.196i)3-s + (−0.316 + 0.316i)5-s − 0.716i·7-s − 0.922i·9-s + (−0.979 + 0.979i)11-s + (0.813 + 0.813i)13-s − 0.124·15-s − 1.94·17-s + (−0.913 − 0.913i)19-s + (0.141 − 0.141i)21-s + 1.17i·23-s − 0.200i·25-s + (0.378 − 0.378i)27-s + (0.259 + 0.259i)29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1929117880\)
\(L(\frac12)\) \(\approx\) \(0.1929117880\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.341 - 0.341i)T + 3iT^{2} \)
7 \( 1 + 1.89iT - 7T^{2} \)
11 \( 1 + (3.24 - 3.24i)T - 11iT^{2} \)
13 \( 1 + (-2.93 - 2.93i)T + 13iT^{2} \)
17 \( 1 + 8.00T + 17T^{2} \)
19 \( 1 + (3.98 + 3.98i)T + 19iT^{2} \)
23 \( 1 - 5.61iT - 23T^{2} \)
29 \( 1 + (-1.39 - 1.39i)T + 29iT^{2} \)
31 \( 1 + 6.29T + 31T^{2} \)
37 \( 1 + (-1.69 + 1.69i)T - 37iT^{2} \)
41 \( 1 - 3.16iT - 41T^{2} \)
43 \( 1 + (3.40 - 3.40i)T - 43iT^{2} \)
47 \( 1 + 5.99T + 47T^{2} \)
53 \( 1 + (1.42 - 1.42i)T - 53iT^{2} \)
59 \( 1 + (9.70 - 9.70i)T - 59iT^{2} \)
61 \( 1 + (-7.04 - 7.04i)T + 61iT^{2} \)
67 \( 1 + (5.84 + 5.84i)T + 67iT^{2} \)
71 \( 1 + 4.49iT - 71T^{2} \)
73 \( 1 + 9.77iT - 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (4.23 + 4.23i)T + 83iT^{2} \)
89 \( 1 + 5.24iT - 89T^{2} \)
97 \( 1 + 5.34T + 97T^{2} \)
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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

−10.04232355230223799074570718682, −9.165936146826413452533047876013, −8.629259696691968805205061814055, −7.40619520698798925499959626053, −6.90840300588734376038896126560, −6.10813942816613079008549140765, −4.59769254812702438496649172265, −4.18243056757446884060216094216, −3.05433634219673143324026343528, −1.82273777106430022655688428824, 0.07236869434030685239852121257, 1.98767839830609167571485740588, 2.84869247149702527986582702048, 4.08962808344510628722144750709, 5.11872271564072469991511428653, 5.86173678002898125956600330178, 6.76223187757241494559586247373, 8.069089695805844182829145514579, 8.357305118413947102476965817114, 8.908879434622236729851610969122

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