Properties

Degree 1
Conductor 131
Sign $-0.306 + 0.951i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $-0.306 + 0.951i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (28, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 131,\ (0:\ ),\ -0.306 + 0.951i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1639517254 + 0.2250647880i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1639517254 + 0.2250647880i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5649723628 + 0.02991589938i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5649723628 + 0.02991589938i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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