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Properties

Label 1-1580-1580.1419-r1-0-0
Degree $1$
Conductor $1580$
Sign $-0.952 + 0.304i$
Analytic cond. $169.794$
Root an. cond. $169.794$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.996 − 0.0804i)3-s + (0.428 + 0.903i)7-s + (0.987 + 0.160i)9-s + (−0.692 + 0.721i)11-s + (0.919 − 0.391i)13-s + (−0.120 + 0.992i)17-s + (0.845 − 0.534i)19-s + (−0.354 − 0.935i)21-s + (−0.5 + 0.866i)23-s + (−0.970 − 0.239i)27-s + (−0.632 − 0.774i)29-s + (0.200 + 0.979i)31-s + (0.748 − 0.663i)33-s + (0.0402 + 0.999i)37-s + (−0.948 + 0.316i)39-s + ⋯
L(s)  = 1  + (−0.996 − 0.0804i)3-s + (0.428 + 0.903i)7-s + (0.987 + 0.160i)9-s + (−0.692 + 0.721i)11-s + (0.919 − 0.391i)13-s + (−0.120 + 0.992i)17-s + (0.845 − 0.534i)19-s + (−0.354 − 0.935i)21-s + (−0.5 + 0.866i)23-s + (−0.970 − 0.239i)27-s + (−0.632 − 0.774i)29-s + (0.200 + 0.979i)31-s + (0.748 − 0.663i)33-s + (0.0402 + 0.999i)37-s + (−0.948 + 0.316i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $-0.952 + 0.304i$
Analytic conductor: \(169.794\)
Root analytic conductor: \(169.794\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (1419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (1:\ ),\ -0.952 + 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1577085935 + 1.011510073i\)
\(L(\frac12)\) \(\approx\) \(0.1577085935 + 1.011510073i\)
\(L(1)\) \(\approx\) \(0.7680149258 + 0.2380580404i\)
\(L(1)\) \(\approx\) \(0.7680149258 + 0.2380580404i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.996 - 0.0804i)T \)
7 \( 1 + (0.428 + 0.903i)T \)
11 \( 1 + (-0.692 + 0.721i)T \)
13 \( 1 + (0.919 - 0.391i)T \)
17 \( 1 + (-0.120 + 0.992i)T \)
19 \( 1 + (0.845 - 0.534i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.632 - 0.774i)T \)
31 \( 1 + (0.200 + 0.979i)T \)
37 \( 1 + (0.0402 + 0.999i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (-0.0402 + 0.999i)T \)
53 \( 1 + (0.996 - 0.0804i)T \)
59 \( 1 + (-0.799 + 0.600i)T \)
61 \( 1 + (0.885 + 0.464i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (-0.568 - 0.822i)T \)
73 \( 1 + (0.919 + 0.391i)T \)
83 \( 1 + (0.799 + 0.600i)T \)
89 \( 1 + (0.568 - 0.822i)T \)
97 \( 1 + (-0.885 - 0.464i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.38496265462733668328976329263, −18.88942212533627712577719738322, −18.417041823377145686165484405134, −17.83395848142210005146293116106, −16.856666395532931897851768284322, −16.273947088873660994578230630267, −15.88556309552997591907863181563, −14.70179785552474933964038823193, −13.71513557676119474290002166687, −13.369993820909859634931611529115, −12.2377713160793566235802013305, −11.497211861173395610164612157831, −10.86278552164135394639460106710, −10.32107566776533243324252598451, −9.36912782340547661720064046489, −8.30725473423289847255162164415, −7.43739288015746373087233117971, −6.765830356999769996821235708980, −5.75693327652732450793669827431, −5.17747175379893657681661640713, −4.17268486724787321527686528899, −3.49789329735345623968610610565, −2.05201900678923239058435403857, −0.93259166602824902043173019813, −0.27451997738006827924793869042, 1.16640388670481116533650699525, 1.92141443178391089057779401549, 3.11066450070565137866664428879, 4.28103379599471086882202185907, 5.12941102027269956107318429769, 5.74677756158610539240810510768, 6.45507233819937539341991268047, 7.55175265284730558156996649588, 8.178173821555743425929667210477, 9.248565086324395634008143444042, 10.10874339555770297113067075605, 10.84859572573492770953545178242, 11.60112239630005950049794483763, 12.20609157907464270432067654799, 13.02679329722371526812614445951, 13.635453857371177563370537835781, 14.98806696540962154525378337251, 15.50426870476940033680286389829, 16.02656393290588682490385858388, 17.08984865972939832227710565560, 17.89747177750045066304582750064, 18.100318638537312063672138389436, 18.95793233115757358643650608007, 19.88615150438165116140276754410, 20.93402024730522105490657457818

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