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Properties

Label 2-640-40.29-c1-0-12
Degree $2$
Conductor $640$
Sign $i$
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  − 2.23·5-s + 2.82i·7-s − 3·9-s − 6.32i·11-s + 4.47·13-s − 6.32i·19-s − 8.48i·23-s + 5.00·25-s − 6.32i·35-s − 4.47·37-s − 2·41-s + 6.70·45-s + 2.82i·47-s − 1.00·49-s − 13.4·53-s + ⋯
L(s)  = 1  − 0.999·5-s + 1.06i·7-s − 9-s − 1.90i·11-s + 1.24·13-s − 1.45i·19-s − 1.76i·23-s + 1.00·25-s − 1.06i·35-s − 0.735·37-s − 0.312·41-s + 0.999·45-s + 0.412i·47-s − 0.142·49-s − 1.84·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599874 - 0.599874i\)
\(L(\frac12)\) \(\approx\) \(0.599874 - 0.599874i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
good3 \( 1 + 3T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 6.32iT - 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6.32iT - 19T^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + 6.32iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.87677590729272902680585206631, −8.987276687909529237611727771945, −8.636346673265982439938560330092, −8.126692865681637123642505443283, −6.56447127430724431282716170581, −5.92594989778316394919276623857, −4.86821913666920457197057313589, −3.45068251819807070971461564270, −2.77121120543910301898558455563, −0.48010892409400119125957988672, 1.54201255175378457978122028951, 3.43758080465563911755674165642, 4.05811579488334999495447688792, 5.20501338902674427373497473276, 6.45824831159538789471893326072, 7.48068970978746736046319225016, 7.925993664530432730077204490368, 9.017292969029582061754595751141, 10.05446416625462425320495466029, 10.79771943209720552012860610333

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