↑

Properties

Label 1-1580-1580.519-r1-0-0
Degree $1$
Conductor $1580$
Sign $0.677 - 0.735i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.428 − 0.903i)3-s + (−0.996 + 0.0804i)7-s + (−0.632 − 0.774i)9-s + (−0.278 + 0.960i)11-s + (−0.799 + 0.600i)13-s + (−0.120 − 0.992i)17-s + (0.0402 − 0.999i)19-s + (−0.354 + 0.935i)21-s + (−0.5 + 0.866i)23-s + (−0.970 + 0.239i)27-s + (0.987 + 0.160i)29-s + (−0.948 + 0.316i)31-s + (0.748 + 0.663i)33-s + (0.845 + 0.534i)37-s + (0.200 + 0.979i)39-s + ⋯
L(s)  = 1  + (0.428 − 0.903i)3-s + (−0.996 + 0.0804i)7-s + (−0.632 − 0.774i)9-s + (−0.278 + 0.960i)11-s + (−0.799 + 0.600i)13-s + (−0.120 − 0.992i)17-s + (0.0402 − 0.999i)19-s + (−0.354 + 0.935i)21-s + (−0.5 + 0.866i)23-s + (−0.970 + 0.239i)27-s + (0.987 + 0.160i)29-s + (−0.948 + 0.316i)31-s + (0.748 + 0.663i)33-s + (0.845 + 0.534i)37-s + (0.200 + 0.979i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(169.794\)
Root analytic conductor: \(169.794\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (1:\ ),\ 0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158687342 - 0.5082920676i\)
\(L(\frac12)\) \(\approx\) \(1.158687342 - 0.5082920676i\)
\(L(1)\) \(\approx\) \(0.8775231686 - 0.2237777920i\)
\(L(1)\) \(\approx\) \(0.8775231686 - 0.2237777920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.428 - 0.903i)T \)
7 \( 1 + (-0.996 + 0.0804i)T \)
11 \( 1 + (-0.278 + 0.960i)T \)
13 \( 1 + (-0.799 + 0.600i)T \)
17 \( 1 + (-0.120 - 0.992i)T \)
19 \( 1 + (0.0402 - 0.999i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.987 + 0.160i)T \)
31 \( 1 + (-0.948 + 0.316i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (-0.428 - 0.903i)T \)
59 \( 1 + (0.919 - 0.391i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (-0.799 - 0.600i)T \)
83 \( 1 + (-0.919 - 0.391i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (-0.885 + 0.464i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.34848762626944593591916851649, −19.71964571412087424169350260339, −19.146445112377510880958682427623, −18.31753596625170772749755007792, −17.144406826231355097235673527503, −16.50981281066280887059656435810, −16.04928107757785839057386194141, −15.14481736226453143162570915270, −14.5431739829774886520783980376, −13.70928666306951751808852329513, −12.92016816548234370608756851328, −12.17463944483320040770291661086, −11.060301841153852295610159603198, −10.14582824802544121157002149119, −10.05214345130057416511212044624, −8.82102876936460303641091718413, −8.303107240757688928710114089501, −7.381034764118755800590732349727, −6.113531097014084696204639403889, −5.650411639117963074658978790573, −4.49913837205009660280668244408, −3.65671976178272542609997289855, −3.02747636006381252463114317923, −2.12006798373726619359741129721, −0.46106311475698529191213286208, 0.42833988040544817662094949771, 1.696581250990791160465571737019, 2.58032638269166581005728370072, 3.19309840475045307574744134348, 4.44213787817728802358333202604, 5.35012045444531131061534907359, 6.52825965102662847692876936827, 7.00588669710461300183858867770, 7.62779991479508878761766634805, 8.72244580177378026850605897358, 9.553551411179490937091126203380, 9.92058865170334189109739028567, 11.406659186763672526211731797580, 11.959872003379237880167409352386, 12.830211401732236134306279981461, 13.26117044315565251118397248494, 14.149013520911443489270988752275, 14.85230172447625539422873589999, 15.742982107959766174118393241590, 16.417413042434853510370530072070, 17.57418216290944640387059589555, 17.90217633389046009727300307983, 18.851367572174524362536490363902, 19.53745922947302135780332590039, 19.98100311162925521500999506993

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