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Properties

Label 1-531-531.115-r1-0-0
Degree $1$
Conductor $531$
Sign $0.575 - 0.817i$
Analytic cond. $57.0638$
Root an. cond. $57.0638$
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  + (0.983 + 0.179i)2-s + (0.935 + 0.353i)4-s + (−0.999 − 0.0361i)5-s + (0.403 + 0.915i)7-s + (0.856 + 0.515i)8-s + (−0.976 − 0.214i)10-s + (−0.197 − 0.980i)11-s + (−0.700 − 0.713i)13-s + (0.232 + 0.972i)14-s + (0.750 + 0.661i)16-s + (−0.994 + 0.108i)17-s + (0.0541 − 0.998i)19-s + (−0.922 − 0.386i)20-s + (−0.0180 − 0.999i)22-s + (0.0901 − 0.995i)23-s + ⋯
L(s)  = 1  + (0.983 + 0.179i)2-s + (0.935 + 0.353i)4-s + (−0.999 − 0.0361i)5-s + (0.403 + 0.915i)7-s + (0.856 + 0.515i)8-s + (−0.976 − 0.214i)10-s + (−0.197 − 0.980i)11-s + (−0.700 − 0.713i)13-s + (0.232 + 0.972i)14-s + (0.750 + 0.661i)16-s + (−0.994 + 0.108i)17-s + (0.0541 − 0.998i)19-s + (−0.922 − 0.386i)20-s + (−0.0180 − 0.999i)22-s + (0.0901 − 0.995i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(57.0638\)
Root analytic conductor: \(57.0638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 531,\ (1:\ ),\ 0.575 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.387628071 - 1.238698159i\)
\(L(\frac12)\) \(\approx\) \(2.387628071 - 1.238698159i\)
\(L(1)\) \(\approx\) \(1.620945125 + 0.01217127438i\)
\(L(1)\) \(\approx\) \(1.620945125 + 0.01217127438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.983 + 0.179i)T \)
5 \( 1 + (-0.999 - 0.0361i)T \)
7 \( 1 + (0.403 + 0.915i)T \)
11 \( 1 + (-0.197 - 0.980i)T \)
13 \( 1 + (-0.700 - 0.713i)T \)
17 \( 1 + (-0.994 + 0.108i)T \)
19 \( 1 + (0.0541 - 0.998i)T \)
23 \( 1 + (0.0901 - 0.995i)T \)
29 \( 1 + (0.336 - 0.941i)T \)
31 \( 1 + (0.891 + 0.452i)T \)
37 \( 1 + (0.856 - 0.515i)T \)
41 \( 1 + (-0.817 + 0.576i)T \)
43 \( 1 + (-0.750 - 0.661i)T \)
47 \( 1 + (0.999 - 0.0361i)T \)
53 \( 1 + (0.976 - 0.214i)T \)
61 \( 1 + (0.983 + 0.179i)T \)
67 \( 1 + (-0.874 + 0.484i)T \)
71 \( 1 + (0.468 - 0.883i)T \)
73 \( 1 + (0.725 - 0.687i)T \)
79 \( 1 + (-0.922 - 0.386i)T \)
83 \( 1 + (0.619 - 0.785i)T \)
89 \( 1 + (-0.647 - 0.762i)T \)
97 \( 1 + (0.232 - 0.972i)T \)
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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

−23.40137219100687037287801619661, −22.71564648439048790831350029424, −21.83025337711559425544434442484, −20.79552950990074661144756388947, −20.08886149244393135556703733125, −19.619703447185516179715088500353, −18.531242209602695197521563010747, −17.252986102750882864902907618260, −16.45063841597489755393292391147, −15.48375641595870314550081694284, −14.8403247196355098668217549660, −14.01103441563265102616139981619, −13.09870384620646649698624649903, −12.10945084825808870204082931636, −11.537933383324705989981518351820, −10.61126722846709533938923424186, −9.73108041553573712656102851295, −8.12348341127297209696660509327, −7.24919206966624032441138487540, −6.72198076482059075584995879488, −5.111055412805302377884444563349, −4.38774498947910516522818266280, −3.7374108441103320396146460636, −2.407008399185552728947514124205, −1.22301589849152711698194210964, 0.487416215706901707933094857142, 2.40186093083155970954980924574, 3.051299864000931588676122918687, 4.35974452888293601251160814727, 5.03854379519246666957545250395, 6.09286394474442449583695638528, 7.08614665757769590876923529427, 8.16251050303556230134701894752, 8.70575814567299871561891326826, 10.46958986257210052669064667465, 11.36198725029910290632401418068, 11.92003133091029651487380727851, 12.813928892098583377201800889516, 13.6680451678235298432229437509, 14.8446019921763445515740774979, 15.33408193670431073827026481944, 15.9815045249148289588907818774, 16.98632416936193773677310529623, 18.07377768849573119265128156985, 19.19731150654712143205607120937, 19.84969259949999591773739310982, 20.73713243954238856579666634431, 21.75797437345450119530535843917, 22.2093732690323795590355927312, 23.136723827940523896787867857697

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