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Properties

Label 1-131-131.27-r0-0-0
Degree $1$
Conductor $131$
Sign $0.993 - 0.111i$
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  + (−0.527 − 0.849i)2-s + (−0.681 − 0.732i)3-s + (−0.443 + 0.896i)4-s + (−0.906 + 0.421i)5-s + (−0.262 + 0.964i)6-s + (−0.998 − 0.0483i)7-s + (0.995 − 0.0965i)8-s + (−0.0724 + 0.997i)9-s + (0.836 + 0.548i)10-s + (0.958 + 0.285i)11-s + (0.958 − 0.285i)12-s + (0.836 − 0.548i)13-s + (0.485 + 0.873i)14-s + (0.926 + 0.377i)15-s + (−0.607 − 0.794i)16-s + (−0.943 + 0.331i)17-s + ⋯
L(s)  = 1  + (−0.527 − 0.849i)2-s + (−0.681 − 0.732i)3-s + (−0.443 + 0.896i)4-s + (−0.906 + 0.421i)5-s + (−0.262 + 0.964i)6-s + (−0.998 − 0.0483i)7-s + (0.995 − 0.0965i)8-s + (−0.0724 + 0.997i)9-s + (0.836 + 0.548i)10-s + (0.958 + 0.285i)11-s + (0.958 − 0.285i)12-s + (0.836 − 0.548i)13-s + (0.485 + 0.873i)14-s + (0.926 + 0.377i)15-s + (−0.607 − 0.794i)16-s + (−0.943 + 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4278841087 + 0.02391630655i\)
\(L(\frac12)\) \(\approx\) \(0.4278841087 + 0.02391630655i\)
\(L(1)\) \(\approx\) \(0.4944454268 - 0.1574379404i\)
\(L(1)\) \(\approx\) \(0.4944454268 - 0.1574379404i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.527 - 0.849i)T \)
3 \( 1 + (-0.681 - 0.732i)T \)
5 \( 1 + (-0.906 + 0.421i)T \)
7 \( 1 + (-0.998 - 0.0483i)T \)
11 \( 1 + (0.958 + 0.285i)T \)
13 \( 1 + (0.836 - 0.548i)T \)
17 \( 1 + (-0.943 + 0.331i)T \)
19 \( 1 + (0.568 + 0.822i)T \)
23 \( 1 + (0.215 + 0.976i)T \)
29 \( 1 + (-0.0724 - 0.997i)T \)
31 \( 1 + (0.399 + 0.916i)T \)
37 \( 1 + (0.715 + 0.698i)T \)
41 \( 1 + (-0.607 + 0.794i)T \)
43 \( 1 + (0.981 + 0.192i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.0241 + 0.999i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.981 - 0.192i)T \)
71 \( 1 + (-0.748 + 0.663i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (-0.168 + 0.985i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.779 + 0.626i)T \)
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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

−28.402892888939414685631446884110, −27.65669705621152624464470699966, −26.70743449426032309980973606518, −26.03476045226714231074196618098, −24.61356268804387095086827315463, −23.75571581966769010315851596104, −22.74405309698082292601660478868, −22.16796745253046432401201427435, −20.39292245666346896481552308816, −19.49359138233137110843839338316, −18.40681716159475981891155955376, −17.12768824603528052782993248838, −16.19028816782926745004666524702, −15.87697686078151846089881922361, −14.70595852579618211802933627736, −13.20674776331864833124048650924, −11.71433930332935376078521250411, −10.77998246941948860595989698654, −9.27769127553340842651799901084, −8.82985651975983209847663830447, −6.98505131815753257009453808023, −6.20218088826420016045688765573, −4.748943047216150290941079128855, −3.74149107326241910226753561970, −0.614330732217260126112903508975, 1.22466464322981361717870256690, 3.02746630227844554171005197791, 4.18271180347444006385912916352, 6.24011956168346322191173582623, 7.29873010698415553524756944265, 8.409051349012439968625068817880, 9.87920207927859527134371909950, 11.05035967788346618548588210612, 11.819969154231400609100522495160, 12.73491419501761622424322782499, 13.717601246121332182214420377792, 15.61693120686234779996833275007, 16.64403831791857589366948425741, 17.71991927277105800176303343149, 18.66651317831343075814401285020, 19.493083198208884520639018771416, 20.101604659455434108437545285546, 21.85265764100963189725354049443, 22.776208576790666235082400565074, 23.14763034232129465536471795659, 24.817502340769411750772132261916, 25.805480578932903919063360916273, 26.990307824558960679673057176394, 27.84329762939666099158759706301, 28.70826681730539483131007674997

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