↑

Properties

Label 2-640-80.43-c1-0-9
Degree $2$
Conductor $640$
Sign $0.779 + 0.626i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.692i·3-s + (−2.22 + 0.245i)5-s + (−0.343 + 0.343i)7-s + 2.52·9-s + (−0.843 + 0.843i)11-s + 3.68·13-s + (0.169 + 1.53i)15-s + (0.412 − 0.412i)17-s + (5.37 − 5.37i)19-s + (0.238 + 0.238i)21-s + (−3.08 − 3.08i)23-s + (4.87 − 1.09i)25-s − 3.82i·27-s + (4.22 + 4.22i)29-s − 8.75i·31-s + ⋯
L(s)  = 1  − 0.399i·3-s + (−0.993 + 0.109i)5-s + (−0.129 + 0.129i)7-s + 0.840·9-s + (−0.254 + 0.254i)11-s + 1.02·13-s + (0.0438 + 0.397i)15-s + (0.0999 − 0.0999i)17-s + (1.23 − 1.23i)19-s + (0.0519 + 0.0519i)21-s + (−0.643 − 0.643i)23-s + (0.975 − 0.218i)25-s − 0.735i·27-s + (0.785 + 0.785i)29-s − 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.779 + 0.626i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26655 - 0.445649i\)
\(L(\frac12)\) \(\approx\) \(1.26655 - 0.445649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.245i)T \)
good3 \( 1 + 0.692iT - 3T^{2} \)
7 \( 1 + (0.343 - 0.343i)T - 7iT^{2} \)
11 \( 1 + (0.843 - 0.843i)T - 11iT^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 + (-0.412 + 0.412i)T - 17iT^{2} \)
19 \( 1 + (-5.37 + 5.37i)T - 19iT^{2} \)
23 \( 1 + (3.08 + 3.08i)T + 23iT^{2} \)
29 \( 1 + (-4.22 - 4.22i)T + 29iT^{2} \)
31 \( 1 + 8.75iT - 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 + 2.54iT - 41T^{2} \)
43 \( 1 + 4.30T + 43T^{2} \)
47 \( 1 + (-4.56 - 4.56i)T + 47iT^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + (-7.33 - 7.33i)T + 59iT^{2} \)
61 \( 1 + (-4.81 + 4.81i)T - 61iT^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 2.97T + 71T^{2} \)
73 \( 1 + (6.87 - 6.87i)T - 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 7.15iT - 83T^{2} \)
89 \( 1 - 1.10T + 89T^{2} \)
97 \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.57563442054093680964226711177, −9.610214386727061096757302331196, −8.617460574213758408064127162860, −7.71552323599537583069425059424, −7.09636773094622979374005171034, −6.14460069506613712215035319339, −4.78668195326818265694269137708, −3.90208685740087321358026683876, −2.67941205874353988249235211451, −0.932636710783442659162183758790, 1.25295204341412503357057322233, 3.33488790641518243419017259373, 3.94500653053633247751174947675, 5.02863674014299411310610078799, 6.16421880787673140227658787875, 7.31142328538787524760573616951, 8.010252516494664218113347751574, 8.864806436291424143850990803700, 10.00465650193807422631442550306, 10.50982336324277871031259329926

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