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Properties

Label 2-640-5.4-c1-0-17
Degree $2$
Conductor $640$
Sign $0.749 + 0.662i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  + 0.806i·3-s + (1.48 − 1.67i)5-s − 2.15i·7-s + 2.35·9-s − 0.387·11-s − 0.962i·13-s + (1.35 + 1.19i)15-s − 1.61i·17-s − 6.31·19-s + 1.73·21-s − 6.15i·23-s + (−0.612 − 4.96i)25-s + 4.31i·27-s + 2·29-s + 9.92·31-s + ⋯
L(s)  = 1  + 0.465i·3-s + (0.662 − 0.749i)5-s − 0.815i·7-s + 0.783·9-s − 0.116·11-s − 0.266i·13-s + (0.348 + 0.308i)15-s − 0.390i·17-s − 1.44·19-s + 0.379·21-s − 1.28i·23-s + (−0.122 − 0.992i)25-s + 0.829i·27-s + 0.371·29-s + 1.78·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57643 - 0.597004i\)
\(L(\frac12)\) \(\approx\) \(1.57643 - 0.597004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 1.67i)T \)
good3 \( 1 - 0.806iT - 3T^{2} \)
7 \( 1 + 2.15iT - 7T^{2} \)
11 \( 1 + 0.387T + 11T^{2} \)
13 \( 1 + 0.962iT - 13T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 + 6.31T + 19T^{2} \)
23 \( 1 + 6.15iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 9.92T + 31T^{2} \)
37 \( 1 + 6.57iT - 37T^{2} \)
41 \( 1 - 4.57T + 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 - 4.54iT - 47T^{2} \)
53 \( 1 - 8.96iT - 53T^{2} \)
59 \( 1 - 6.31T + 59T^{2} \)
61 \( 1 + 0.261T + 61T^{2} \)
67 \( 1 + 9.89iT - 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 13.0iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 - 2.88iT - 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 9.61iT - 97T^{2} \)
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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

−10.33681573057457494287778620008, −9.774321625232747648648124251460, −8.820073820507523504152890342893, −7.968396880596562750781051849881, −6.83078597963841665227151008462, −5.95957945180450222158766966506, −4.55237674806378630707577898301, −4.33600595306399203347973412167, −2.56428078179638889180096969689, −1.00425900612344803042979019991, 1.72016547242592126433201842531, 2.64154007109568605618910121346, 4.06654860351470032522119982828, 5.38840943291143522159061128610, 6.35631955553842477272989105718, 6.91688971137542329499775535452, 8.037704741721134383627781553532, 8.935139813918427607626676174560, 9.971011401977400296015727878674, 10.47939455713943616910508459197

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