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Properties

Label 2-640-80.69-c1-0-9
Degree $2$
Conductor $640$
Sign $0.962 + 0.269i$
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  + (0.734 + 0.734i)3-s + (1.17 − 1.90i)5-s − 1.71·7-s − 1.92i·9-s + (2.82 + 2.82i)11-s + (2.59 + 2.59i)13-s + (2.25 − 0.537i)15-s − 1.89i·17-s + (2.89 − 2.89i)19-s + (−1.25 − 1.25i)21-s − 2.00·23-s + (−2.25 − 4.46i)25-s + (3.61 − 3.61i)27-s + (6.72 − 6.72i)29-s + 7.11·31-s + ⋯
L(s)  = 1  + (0.423 + 0.423i)3-s + (0.524 − 0.851i)5-s − 0.648·7-s − 0.640i·9-s + (0.852 + 0.852i)11-s + (0.719 + 0.719i)13-s + (0.583 − 0.138i)15-s − 0.460i·17-s + (0.664 − 0.664i)19-s + (−0.274 − 0.274i)21-s − 0.418·23-s + (−0.450 − 0.892i)25-s + (0.695 − 0.695i)27-s + (1.24 − 1.24i)29-s + 1.27·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.962 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86424 - 0.256276i\)
\(L(\frac12)\) \(\approx\) \(1.86424 - 0.256276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 + 1.90i)T \)
good3 \( 1 + (-0.734 - 0.734i)T + 3iT^{2} \)
7 \( 1 + 1.71T + 7T^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \)
13 \( 1 + (-2.59 - 2.59i)T + 13iT^{2} \)
17 \( 1 + 1.89iT - 17T^{2} \)
19 \( 1 + (-2.89 + 2.89i)T - 19iT^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + (-6.72 + 6.72i)T - 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (2.25 - 2.25i)T - 37iT^{2} \)
41 \( 1 + 1.59iT - 41T^{2} \)
43 \( 1 + (8.06 - 8.06i)T - 43iT^{2} \)
47 \( 1 - 4.43iT - 47T^{2} \)
53 \( 1 + (0.481 - 0.481i)T - 53iT^{2} \)
59 \( 1 + (-3.08 - 3.08i)T + 59iT^{2} \)
61 \( 1 + (3.46 - 3.46i)T - 61iT^{2} \)
67 \( 1 + (-1.80 - 1.80i)T + 67iT^{2} \)
71 \( 1 - 0.379iT - 71T^{2} \)
73 \( 1 - 8.37T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + (8.24 + 8.24i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + 6.50iT - 97T^{2} \)
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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

−10.03840352500267674947322762472, −9.642406990301583284221286587166, −9.027699478993014764578491701123, −8.195497623215483950024078918414, −6.70628851974900006470182907542, −6.23333262452077610352988101229, −4.75996783757223605727182137978, −4.06763464440525479449552784847, −2.78925011207558333786063856349, −1.19527990505765903073388635666, 1.50272167740527704846676323283, 2.93946786300850675042583784199, 3.59889725486319411217060240965, 5.36084934990672785126573454213, 6.29447781012889912669341638745, 6.91128452867199976197045952403, 8.103816430685921229214635700366, 8.696857670881043836998366098918, 9.934619550379517745383522223133, 10.47323289022203372569769828884

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