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Properties

Label 2-640-40.37-c0-0-1
Degree $2$
Conductor $640$
Sign $0.525 + 0.850i$
Analytic cond. $0.319401$
Root an. cond. $0.565156$
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  − i·5-s − i·9-s + (−1 − i)13-s + (1 + i)17-s − 25-s + 2·29-s + (−1 + i)37-s − 45-s − i·49-s + (1 + i)53-s + 2i·61-s + (−1 + i)65-s + (−1 + i)73-s − 81-s + (1 − i)85-s + ⋯
L(s)  = 1  − i·5-s − i·9-s + (−1 − i)13-s + (1 + i)17-s − 25-s + 2·29-s + (−1 + i)37-s − 45-s − i·49-s + (1 + i)53-s + 2i·61-s + (−1 + i)65-s + (−1 + i)73-s − 81-s + (1 − i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8996836088\)
\(L(\frac12)\) \(\approx\) \(0.8996836088\)
\(L(1)\) 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 \( 1 + iT \)
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 \( 1 - 2T + T^{2} \)
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 \( 1 - 2iT - T^{2} \)
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

−10.27210261224334951707741205924, −9.965230818087188628009661043267, −8.771096803156510626858694076701, −8.225542819045552508889012179479, −7.15759060544865267543629220444, −5.99998328598075652176813193167, −5.19175679678103294419451207264, −4.12479089167586388659587382825, −2.94619475601130156348403515238, −1.13948448043914854571103288946, 2.13757936438182458151518482210, 3.09481422652681407443936502056, 4.50656513587056920479614836398, 5.41313518142831421682778490031, 6.68569085601186223482794025882, 7.31716713927982669533484695021, 8.136901511990807971689526809894, 9.389510267281642706992122299767, 10.13056497054052107227578354334, 10.82958733702769061113364639761

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