↑

Properties

Label 1-500-500.31-r1-0-0
Degree $1$
Conductor $500$
Sign $0.597 - 0.801i$
Analytic cond. $53.7324$
Root an. cond. $53.7324$
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  + (0.637 − 0.770i)3-s + (−0.309 + 0.951i)7-s + (−0.187 − 0.982i)9-s + (0.992 − 0.125i)11-s + (−0.187 − 0.982i)13-s + (0.968 + 0.248i)17-s + (0.637 + 0.770i)19-s + (0.535 + 0.844i)21-s + (−0.728 + 0.684i)23-s + (−0.876 − 0.481i)27-s + (0.0627 − 0.998i)29-s + (−0.968 − 0.248i)31-s + (0.535 − 0.844i)33-s + (0.876 − 0.481i)37-s + (−0.876 − 0.481i)39-s + ⋯
L(s)  = 1  + (0.637 − 0.770i)3-s + (−0.309 + 0.951i)7-s + (−0.187 − 0.982i)9-s + (0.992 − 0.125i)11-s + (−0.187 − 0.982i)13-s + (0.968 + 0.248i)17-s + (0.637 + 0.770i)19-s + (0.535 + 0.844i)21-s + (−0.728 + 0.684i)23-s + (−0.876 − 0.481i)27-s + (0.0627 − 0.998i)29-s + (−0.968 − 0.248i)31-s + (0.535 − 0.844i)33-s + (0.876 − 0.481i)37-s + (−0.876 − 0.481i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.597 - 0.801i$
Analytic conductor: \(53.7324\)
Root analytic conductor: \(53.7324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 500,\ (1:\ ),\ 0.597 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.417507468 - 1.212709262i\)
\(L(\frac12)\) \(\approx\) \(2.417507468 - 1.212709262i\)
\(L(1)\) \(\approx\) \(1.386252183 - 0.3455781685i\)
\(L(1)\) \(\approx\) \(1.386252183 - 0.3455781685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.637 - 0.770i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.992 - 0.125i)T \)
13 \( 1 + (-0.187 - 0.982i)T \)
17 \( 1 + (0.968 + 0.248i)T \)
19 \( 1 + (0.637 + 0.770i)T \)
23 \( 1 + (-0.728 + 0.684i)T \)
29 \( 1 + (0.0627 - 0.998i)T \)
31 \( 1 + (-0.968 - 0.248i)T \)
37 \( 1 + (0.876 - 0.481i)T \)
41 \( 1 + (0.728 + 0.684i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (0.929 + 0.368i)T \)
53 \( 1 + (0.535 + 0.844i)T \)
59 \( 1 + (0.425 - 0.904i)T \)
61 \( 1 + (0.728 - 0.684i)T \)
67 \( 1 + (-0.0627 - 0.998i)T \)
71 \( 1 + (0.929 + 0.368i)T \)
73 \( 1 + (-0.425 - 0.904i)T \)
79 \( 1 + (0.637 - 0.770i)T \)
83 \( 1 + (0.637 + 0.770i)T \)
89 \( 1 + (-0.425 - 0.904i)T \)
97 \( 1 + (0.0627 - 0.998i)T \)
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

−23.61683343324507752265230521439, −22.46003434108911122883012515786, −21.97341384366462094973473522563, −20.94585881557502052231994655823, −20.12842826658444206770448750057, −19.634022338041796215406875817953, −18.68698853299521196881103611822, −17.405503088834927185563324480055, −16.43645109805982565911635812142, −16.18250219141709735883822076911, −14.72254676884099242419448073167, −14.24426471643735489671124889079, −13.51657203895886616887752844604, −12.25230012542899241626879244544, −11.23164026688846460905938319988, −10.28332782533274568828626716542, −9.48723506347021098932358946020, −8.810535740147379191915176643461, −7.51460233547545238411125516084, −6.81211647358385085976303022089, −5.3710060591127246647110636016, −4.21079934291236075345054858853, −3.686674990486427018981861548989, −2.42787693554396137726887231732, −1.01286928461070645121849642836, 0.78514633131259227572918275274, 1.94039163267154252779633788851, 3.02735280071706479574606856915, 3.866787275166784594504145098060, 5.69421120126083458826095181738, 6.121155135076055750332580419006, 7.5466038055758010304741165360, 8.09063213701102495354441923440, 9.26736189163795498526878409116, 9.7973990228841283047560289458, 11.38945340483145782467338061668, 12.26514624487826742995816335765, 12.75212142648091087162982981802, 13.93521470166725813570783089591, 14.63310535144756449236645391448, 15.41450106463073258866711132593, 16.49208949030411967767154664652, 17.60075951276982059326223558004, 18.32926569586882295013516493843, 19.14294351924745961321885224590, 19.78628002289850697650219832658, 20.64555560369356766059464001608, 21.65962119054041041548278166508, 22.52007175887182191161379883774, 23.35092209595671851448292293046

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