Properties

Label 2-1280-5.2-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.234704274\)
\(L(\frac12)\) \(\approx\) \(1.234704274\)
\(L(1)\) 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

−9.964832605681071972461891305552, −9.291224714006104767552104887300, −8.291711220449639671757808819552, −7.52536394127897905866697190257, −6.59235801628521209496711360420, −5.70376606193782252549884416415, −4.99125058016544368812370535309, −3.96212254881160499698511916202, −2.48842071831408910394253778505, −1.77841542129944846033303856091, 1.20109791377981409238931281914, 2.70701320733826131018143723953, 3.43923422826357538745900098234, 5.05250576282773688095770727869, 5.41866129202010967020773057736, 6.57314167631080201188453586388, 7.17438654156441139716822791752, 8.248164377289671937300698850356, 9.186141973908284260333454055670, 9.799743619951564699777668367968

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