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Properties

Label 2-640-80.67-c1-0-2
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

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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: $\chi_{640} (607, \cdot )$
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 \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \)
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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

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

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