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Properties

Label 2-160-5.4-c1-0-4
Degree $2$
Conductor $160$
Sign $i$
Analytic cond. $1.27760$
Root an. cond. $1.13031$
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  − 1.23i·3-s − 2.23·5-s − 5.23i·7-s + 1.47·9-s + 2.76i·15-s − 6.47·21-s + 7.70i·23-s + 5.00·25-s − 5.52i·27-s + 6·29-s + 11.7i·35-s + 4.47·41-s + 6.76i·43-s − 3.29·45-s − 0.291i·47-s + ⋯
L(s)  = 1  − 0.713i·3-s − 0.999·5-s − 1.97i·7-s + 0.490·9-s + 0.713i·15-s − 1.41·21-s + 1.60i·23-s + 1.00·25-s − 1.06i·27-s + 1.11·29-s + 1.97i·35-s + 0.698·41-s + 1.03i·43-s − 0.490·45-s − 0.0425i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688907 - 0.688907i\)
\(L(\frac12)\) \(\approx\) \(0.688907 - 0.688907i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.23iT - 3T^{2} \)
7 \( 1 + 5.23iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 7.70iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 - 6.76iT - 43T^{2} \)
47 \( 1 + 0.291iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 14.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.29iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
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

−12.76892970195230386550627255190, −11.64665269177327489585132114264, −10.75114764759953690220526980979, −9.761651929730735676204324529262, −8.023666956296543312925465170693, −7.42541760646773038739521629905, −6.65989392962173684406985324141, −4.58001151796818634463342727799, −3.60042315231287594743817683746, −1.05253035425592649295608201792, 2.71110513399110985174563244987, 4.24641408342865867467795153238, 5.33588751391846393555097535154, 6.74347725118726718629208840651, 8.290531743834660130877881971853, 8.931235541128439524930930999071, 10.10100362009494236441728890724, 11.23121402096295023866785027521, 12.18815404261445079302592801273, 12.72213446674387754307118090423

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