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Properties

Label 2-1280-40.37-c0-0-1
Degree $2$
Conductor $1280$
Sign $-0.229 + 0.973i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6285584519\)
\(L(\frac12)\) \(\approx\) \(0.6285584519\)
\(L(1)\) 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

−9.549043106157519284353947974808, −8.862133571389173939098725474424, −7.925364502967568629946506837545, −7.23804426260962121104767848944, −6.49651926228537385469881830752, −5.30401274665158675155824955763, −4.45135557577723848392513528084, −3.48790581430571638952733573985, −2.55676696881410933695317694536, −0.52541391397976017058311200122, 1.86337020966777733282503932303, 2.99792808947332476694819824521, 4.43813509290071215277046417838, 4.58972446652015617176234769491, 6.01829734414518287410648994780, 6.98510380780720997632179167339, 7.67555328746532838095396762153, 8.390080089110390140436706790188, 9.194119156098533152783828734163, 10.17838798312829118638460205802

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