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Properties

Label 1-131-131.38-r0-0-0
Degree $1$
Conductor $131$
Sign $0.898 - 0.439i$
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  + (−0.168 + 0.985i)2-s + (0.926 − 0.377i)3-s + (−0.943 − 0.331i)4-s + (−0.0724 − 0.997i)5-s + (0.215 + 0.976i)6-s + (−0.861 − 0.506i)7-s + (0.485 − 0.873i)8-s + (0.715 − 0.698i)9-s + (0.995 + 0.0965i)10-s + (−0.998 − 0.0483i)11-s + (−0.998 + 0.0483i)12-s + (0.995 − 0.0965i)13-s + (0.644 − 0.764i)14-s + (−0.443 − 0.896i)15-s + (0.779 + 0.626i)16-s + (0.836 − 0.548i)17-s + ⋯
L(s)  = 1  + (−0.168 + 0.985i)2-s + (0.926 − 0.377i)3-s + (−0.943 − 0.331i)4-s + (−0.0724 − 0.997i)5-s + (0.215 + 0.976i)6-s + (−0.861 − 0.506i)7-s + (0.485 − 0.873i)8-s + (0.715 − 0.698i)9-s + (0.995 + 0.0965i)10-s + (−0.998 − 0.0483i)11-s + (−0.998 + 0.0483i)12-s + (0.995 − 0.0965i)13-s + (0.644 − 0.764i)14-s + (−0.443 − 0.896i)15-s + (0.779 + 0.626i)16-s + (0.836 − 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.898 - 0.439i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.898 - 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034219650 - 0.2394209242i\)
\(L(\frac12)\) \(\approx\) \(1.034219650 - 0.2394209242i\)
\(L(1)\) \(\approx\) \(1.060145920 + 0.0004807760724i\)
\(L(1)\) \(\approx\) \(1.060145920 + 0.0004807760724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.168 + 0.985i)T \)
3 \( 1 + (0.926 - 0.377i)T \)
5 \( 1 + (-0.0724 - 0.997i)T \)
7 \( 1 + (-0.861 - 0.506i)T \)
11 \( 1 + (-0.998 - 0.0483i)T \)
13 \( 1 + (0.995 - 0.0965i)T \)
17 \( 1 + (0.836 - 0.548i)T \)
19 \( 1 + (-0.354 - 0.935i)T \)
23 \( 1 + (-0.681 + 0.732i)T \)
29 \( 1 + (0.715 + 0.698i)T \)
31 \( 1 + (0.981 + 0.192i)T \)
37 \( 1 + (-0.607 + 0.794i)T \)
41 \( 1 + (0.779 - 0.626i)T \)
43 \( 1 + (-0.527 + 0.849i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.262 - 0.964i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.527 - 0.849i)T \)
71 \( 1 + (0.120 + 0.992i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (0.958 + 0.285i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.399 + 0.916i)T \)
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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

−28.71756842313094289064646401538, −27.841266890959620743933620867392, −26.600386502881581164956918281519, −26.08387686286558313360253283979, −25.302660436070756753592619908023, −23.33085323984731206063212727653, −22.53883279648350205341428525890, −21.39929185867863556924271676755, −20.8562240298319149606813225529, −19.486052331965765410438799562517, −18.85247216357514561397268731037, −18.180331570775163911881619913431, −16.32627554412748730771995983490, −15.24894014711020827221313304246, −14.11409126930906334840558465696, −13.21236905659272838908622556577, −12.06394768545018766849625878806, −10.3747502792924435417884031777, −10.19482711141102084971268454902, −8.70742722208499121886964083813, −7.78970835646258841906818075773, −5.950816229442977197295649712502, −4.02431361902891891399867011429, −3.11721457178432953990150350227, −2.18851987689614023478865865651, 1.00652771392818414722411978895, 3.2783779724224561312750676583, 4.61918035135654122870386062100, 6.070766857421256388260489418945, 7.36259893600202319927514387692, 8.26819034407800645243868207749, 9.19490615441808783205810435670, 10.19957043436310813027594556044, 12.494653804689745602770896184, 13.37908507813202719890974602644, 13.91972013517293116740681700245, 15.70311108589455752842304350923, 15.904128184194948810810153578446, 17.30376476861604078861846951090, 18.44248815805824769750784712475, 19.408525737029945782651100254010, 20.36709661291322916398416243651, 21.46177130059565141213115853969, 23.2517446449759526108423262798, 23.6719640632281962550313062506, 24.76803538648357167826631953723, 25.7442278257129860285640081488, 26.13299512749336229426959711634, 27.42965098032201940520964401948, 28.43492599893481764101297690839

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