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Properties

Label 1-131-131.28-r0-0-0
Degree $1$
Conductor $131$
Sign $-0.306 + 0.951i$
Analytic cond. $0.608361$
Root an. cond. $0.608361$
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  + (0.0241 − 0.999i)2-s + (−0.168 + 0.985i)3-s + (−0.998 − 0.0483i)4-s + (−0.443 − 0.896i)5-s + (0.981 + 0.192i)6-s + (−0.681 + 0.732i)7-s + (−0.0724 + 0.997i)8-s + (−0.943 − 0.331i)9-s + (−0.906 + 0.421i)10-s + (0.215 + 0.976i)11-s + (0.215 − 0.976i)12-s + (−0.906 − 0.421i)13-s + (0.715 + 0.698i)14-s + (0.958 − 0.285i)15-s + (0.995 + 0.0965i)16-s + (−0.861 + 0.506i)17-s + ⋯
L(s)  = 1  + (0.0241 − 0.999i)2-s + (−0.168 + 0.985i)3-s + (−0.998 − 0.0483i)4-s + (−0.443 − 0.896i)5-s + (0.981 + 0.192i)6-s + (−0.681 + 0.732i)7-s + (−0.0724 + 0.997i)8-s + (−0.943 − 0.331i)9-s + (−0.906 + 0.421i)10-s + (0.215 + 0.976i)11-s + (0.215 − 0.976i)12-s + (−0.906 − 0.421i)13-s + (0.715 + 0.698i)14-s + (0.958 − 0.285i)15-s + (0.995 + 0.0965i)16-s + (−0.861 + 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $-0.306 + 0.951i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ -0.306 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1639517254 + 0.2250647880i\)
\(L(\frac12)\) \(\approx\) \(0.1639517254 + 0.2250647880i\)
\(L(1)\) \(\approx\) \(0.5649723628 + 0.02991589938i\)
\(L(1)\) \(\approx\) \(0.5649723628 + 0.02991589938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.0241 - 0.999i)T \)
3 \( 1 + (-0.168 + 0.985i)T \)
5 \( 1 + (-0.443 - 0.896i)T \)
7 \( 1 + (-0.681 + 0.732i)T \)
11 \( 1 + (0.215 + 0.976i)T \)
13 \( 1 + (-0.906 - 0.421i)T \)
17 \( 1 + (-0.861 + 0.506i)T \)
19 \( 1 + (-0.748 + 0.663i)T \)
23 \( 1 + (-0.527 + 0.849i)T \)
29 \( 1 + (-0.943 + 0.331i)T \)
31 \( 1 + (0.644 - 0.764i)T \)
37 \( 1 + (0.836 - 0.548i)T \)
41 \( 1 + (0.995 - 0.0965i)T \)
43 \( 1 + (-0.989 + 0.144i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.399 - 0.916i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.989 - 0.144i)T \)
71 \( 1 + (-0.970 + 0.239i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.262 + 0.964i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.485 + 0.873i)T \)
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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

−28.406876166897476095474028798263, −26.80637498363934995887083895337, −26.49031822647963084328547373070, −25.33132323533839507051069705332, −24.24442138955650730057304025299, −23.61618400898064882840755190170, −22.59134905653065532401212303668, −22.001930527689444807029971539234, −19.724618457441227464459254082210, −19.1116967635391261179176501199, −18.164151079850583517775957454243, −17.08365048253022847434725999622, −16.23996900600442813968557320291, −14.86912557694487832239124147882, −13.911556622313534405906183266883, −13.15918645202544231881017683343, −11.778668711102074937399797906878, −10.487811695823793060191150972391, −8.91766488390107447502333782515, −7.65407065098962190617015222395, −6.813159312906350832922868488487, −6.18484248035067143007980219394, −4.352784784019643750915697585735, −2.83249734007650434154491985995, −0.24466023709320682042201371358, 2.21689600840805318364866716957, 3.76747082550298336631659602378, 4.65234245995783497026996482667, 5.77575988345520778760492877426, 8.13407238242039771610620080848, 9.32992114643996639596970046438, 9.82832650898520971680586655385, 11.244327168368162187645364442952, 12.28704730859314958038668072984, 12.92635764008023960517869273708, 14.726796740736366760283769090112, 15.53548649176196311228615261812, 16.84937883610411460181315807334, 17.69764953323487725255541261870, 19.32810104969143036143324945856, 19.992652387404319393906114677707, 20.86520450472762216133317235195, 21.935446590756681159568464605126, 22.57980521316152883261938997683, 23.63328464543423335664143776916, 25.14794376896429393053430720922, 26.31844520163906307966175171165, 27.43183408821744179643285115127, 28.09084055775164168813056202125, 28.65233154735507189488544569849

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