↑

Properties

Label 1-800-800.573-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.389 + 0.920i$
Analytic cond. $85.9719$
Root an. cond. $85.9719$
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  + (0.987 − 0.156i)3-s − 7-s + (0.951 − 0.309i)9-s + (−0.453 − 0.891i)11-s + (0.891 + 0.453i)13-s + (−0.587 − 0.809i)17-s + (−0.987 − 0.156i)19-s + (−0.987 + 0.156i)21-s + (−0.309 + 0.951i)23-s + (0.891 − 0.453i)27-s + (0.156 + 0.987i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.453 + 0.891i)37-s + (0.951 + 0.309i)39-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)3-s − 7-s + (0.951 − 0.309i)9-s + (−0.453 − 0.891i)11-s + (0.891 + 0.453i)13-s + (−0.587 − 0.809i)17-s + (−0.987 − 0.156i)19-s + (−0.987 + 0.156i)21-s + (−0.309 + 0.951i)23-s + (0.891 − 0.453i)27-s + (0.156 + 0.987i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.453 + 0.891i)37-s + (0.951 + 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5948208424 + 0.8978248533i\)
\(L(\frac12)\) \(\approx\) \(0.5948208424 + 0.8978248533i\)
\(L(1)\) \(\approx\) \(1.137203084 + 0.005279880616i\)
\(L(1)\) \(\approx\) \(1.137203084 + 0.005279880616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.987 - 0.156i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.453 - 0.891i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.987 - 0.156i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.453 + 0.891i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.156 - 0.987i)T \)
59 \( 1 + (0.891 + 0.453i)T \)
61 \( 1 + (0.891 - 0.453i)T \)
67 \( 1 + (-0.156 + 0.987i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.156 + 0.987i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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

−21.794350289369869966613632184108, −20.71574116630945218095917639344, −20.38145246632478455316986379499, −19.37558967573440803686483246861, −18.859765313756595581201071206487, −17.92733161398482682835001387796, −16.90692348410677492424398747686, −15.852786740212310683913387135724, −15.4029495165626966321278117587, −14.61595930190212434438908400975, −13.5329008553445663934265870642, −12.924372123024549776338287379, −12.357895385475630336883192727698, −10.67518223582216050038446658098, −10.291333952472210734267598686760, −9.24365103383139444232831247612, −8.55789336900650141749402368120, −7.66363125861865397897347345193, −6.696281876150132123584735669480, −5.82192504825156349348410811571, −4.330169273315097000393784898324, −3.784781407986036065387256489629, −2.62201686421011962287769584434, −1.8753451262980756602130742070, −0.199673697546532159255657381, 1.20991453829272633918126944005, 2.44219644560465554284237315233, 3.31078053213660689104340785707, 3.997260586732262228787737746325, 5.37083219218103257542634909893, 6.54546040948965389255012815530, 7.10054407906913746997529075706, 8.414133887067323658724832329697, 8.84325858533785244649645578385, 9.76222167011534076978312194154, 10.668248311874166890170634652589, 11.66539821516061592073849123370, 12.88137654340774982681208009058, 13.360823677305810094730059522585, 14.01019700785118495628555799386, 15.03529967498047873677350073082, 16.04706644485474259149811970214, 16.1998951701483182636452118657, 17.66825303746583747904085975381, 18.63201881194467057147303514251, 19.06485868016140895122374588079, 19.90231311818423790163134387916, 20.59814679137165070277978381594, 21.5406098921830196027450684620, 22.058777494220974736381945536764

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