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Properties

Label 2-640-5.4-c1-0-7
Degree $2$
Conductor $640$
Sign $0.990 - 0.139i$
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  − 2.90i·3-s + (0.311 + 2.21i)5-s + 3.52i·7-s − 5.42·9-s + 3.80·11-s + 2.62i·13-s + (6.42 − 0.903i)15-s + 5.80i·17-s + 5.05·19-s + 10.2·21-s − 0.474i·23-s + (−4.80 + 1.37i)25-s + 7.05i·27-s − 2·29-s − 2.75·31-s + ⋯
L(s)  = 1  − 1.67i·3-s + (0.139 + 0.990i)5-s + 1.33i·7-s − 1.80·9-s + 1.14·11-s + 0.727i·13-s + (1.65 − 0.233i)15-s + 1.40i·17-s + 1.15·19-s + 2.23·21-s − 0.0989i·23-s + (−0.961 + 0.275i)25-s + 1.35i·27-s − 0.371·29-s − 0.494·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.990 - 0.139i$
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: \((2,\ 640,\ (\ :1/2),\ 0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50215 + 0.105009i\)
\(L(\frac12)\) \(\approx\) \(1.50215 + 0.105009i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.311 - 2.21i)T \)
good3 \( 1 + 2.90iT - 3T^{2} \)
7 \( 1 - 3.52iT - 7T^{2} \)
11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 - 2.62iT - 13T^{2} \)
17 \( 1 - 5.80iT - 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 + 0.474iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
37 \( 1 + 7.18iT - 37T^{2} \)
41 \( 1 - 5.18T + 41T^{2} \)
43 \( 1 - 1.95iT - 43T^{2} \)
47 \( 1 + 5.33iT - 47T^{2} \)
53 \( 1 - 5.37iT - 53T^{2} \)
59 \( 1 + 5.05T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 7.76iT - 67T^{2} \)
71 \( 1 + 4.85T + 71T^{2} \)
73 \( 1 - 6.66iT - 73T^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−10.91100721911104003899166123092, −9.478426027457014519264847065592, −8.783798161513166031194311813743, −7.76842796206476162647302307070, −6.96881599148936617551710609413, −6.21817837838549063003196842060, −5.68284981720284476121226986336, −3.67944701913127203620477790325, −2.41088061620223998540545979992, −1.62078477626854186280736772918, 0.901204714652685715508294568582, 3.26488095340401798146517027395, 4.09564573872738959995880026514, 4.84145966415463868409578408818, 5.62110888284490855133152463288, 7.07614367087467986560794737074, 8.122497301501324793317918745053, 9.295154003574856759065873855141, 9.549598531832885635262398119834, 10.36105697935487621119699950495

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