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Properties

Label 2-640-16.5-c1-0-14
Degree $2$
Conductor $640$
Sign $-0.289 + 0.957i$
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  + (2.32 − 2.32i)3-s + (−0.707 − 0.707i)5-s − 0.982i·7-s − 7.82i·9-s + (1.62 + 1.62i)11-s + (0.690 − 0.690i)13-s − 3.28·15-s − 2.19·17-s + (−1.92 + 1.92i)19-s + (−2.28 − 2.28i)21-s + 2.01i·23-s + 1.00i·25-s + (−11.2 − 11.2i)27-s + (5.27 − 5.27i)29-s + 0.435·31-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)3-s + (−0.316 − 0.316i)5-s − 0.371i·7-s − 2.60i·9-s + (0.490 + 0.490i)11-s + (0.191 − 0.191i)13-s − 0.849·15-s − 0.532·17-s + (−0.441 + 0.441i)19-s + (−0.498 − 0.498i)21-s + 0.420i·23-s + 0.200i·25-s + (−2.15 − 2.15i)27-s + (0.978 − 0.978i)29-s + 0.0781·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.289 + 0.957i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.289 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28560 - 1.73173i\)
\(L(\frac12)\) \(\approx\) \(1.28560 - 1.73173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-2.32 + 2.32i)T - 3iT^{2} \)
7 \( 1 + 0.982iT - 7T^{2} \)
11 \( 1 + (-1.62 - 1.62i)T + 11iT^{2} \)
13 \( 1 + (-0.690 + 0.690i)T - 13iT^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 + (1.92 - 1.92i)T - 19iT^{2} \)
23 \( 1 - 2.01iT - 23T^{2} \)
29 \( 1 + (-5.27 + 5.27i)T - 29iT^{2} \)
31 \( 1 - 0.435T + 31T^{2} \)
37 \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (-0.507 - 0.507i)T + 43iT^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 + (6.29 + 6.29i)T + 53iT^{2} \)
59 \( 1 + (-5.67 - 5.67i)T + 59iT^{2} \)
61 \( 1 + (-3.60 + 3.60i)T - 61iT^{2} \)
67 \( 1 + (4.53 - 4.53i)T - 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 9.24iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (-0.683 + 0.683i)T - 83iT^{2} \)
89 \( 1 - 5.44iT - 89T^{2} \)
97 \( 1 - 5.54T + 97T^{2} \)
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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

−10.02812049256093151542626804030, −9.238199616938810416655220800969, −8.299888884100483642257643100603, −7.88541425824026157240563538152, −6.88860951917370531060059136174, −6.24373597975553617767954599237, −4.44389361113217928998905732908, −3.44749828450000681379527280582, −2.26237850858833854702287852075, −1.09533379243630676924571422233, 2.30266500418928532739067243768, 3.25305661799300791507292592273, 4.13267216020171676629643722907, 4.97439744017763624647563391119, 6.37882477450529087464069076908, 7.59768809178029009897621630488, 8.633370419307247054933543215370, 8.910672125270815743454003430323, 9.849734628041518014877277895049, 10.74723458620086636566070038051

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