↑

Properties

Label 1-131-131.12-r0-0-0
Degree $1$
Conductor $131$
Sign $0.975 - 0.221i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.906 − 0.421i)2-s + (0.995 − 0.0965i)3-s + (0.644 + 0.764i)4-s + (0.399 + 0.916i)5-s + (−0.943 − 0.331i)6-s + (−0.607 − 0.794i)7-s + (−0.262 − 0.964i)8-s + (0.981 − 0.192i)9-s + (0.0241 − 0.999i)10-s + (0.715 − 0.698i)11-s + (0.715 + 0.698i)12-s + (0.0241 + 0.999i)13-s + (0.215 + 0.976i)14-s + (0.485 + 0.873i)15-s + (−0.168 + 0.985i)16-s + (−0.989 + 0.144i)17-s + ⋯
L(s)  = 1  + (−0.906 − 0.421i)2-s + (0.995 − 0.0965i)3-s + (0.644 + 0.764i)4-s + (0.399 + 0.916i)5-s + (−0.943 − 0.331i)6-s + (−0.607 − 0.794i)7-s + (−0.262 − 0.964i)8-s + (0.981 − 0.192i)9-s + (0.0241 − 0.999i)10-s + (0.715 − 0.698i)11-s + (0.715 + 0.698i)12-s + (0.0241 + 0.999i)13-s + (0.215 + 0.976i)14-s + (0.485 + 0.873i)15-s + (−0.168 + 0.985i)16-s + (−0.989 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035819713 - 0.1160563502i\)
\(L(\frac12)\) \(\approx\) \(1.035819713 - 0.1160563502i\)
\(L(1)\) \(\approx\) \(1.002637949 - 0.1052982630i\)
\(L(1)\) \(\approx\) \(1.002637949 - 0.1052982630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.906 - 0.421i)T \)
3 \( 1 + (0.995 - 0.0965i)T \)
5 \( 1 + (0.399 + 0.916i)T \)
7 \( 1 + (-0.607 - 0.794i)T \)
11 \( 1 + (0.715 - 0.698i)T \)
13 \( 1 + (0.0241 + 0.999i)T \)
17 \( 1 + (-0.989 + 0.144i)T \)
19 \( 1 + (0.885 - 0.464i)T \)
23 \( 1 + (0.836 + 0.548i)T \)
29 \( 1 + (0.981 + 0.192i)T \)
31 \( 1 + (-0.998 - 0.0483i)T \)
37 \( 1 + (-0.527 + 0.849i)T \)
41 \( 1 + (-0.168 - 0.985i)T \)
43 \( 1 + (-0.861 - 0.506i)T \)
47 \( 1 + (0.120 - 0.992i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.443 - 0.896i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.861 + 0.506i)T \)
71 \( 1 + (-0.354 + 0.935i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (-0.0724 + 0.997i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.958 + 0.285i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.523711941944094025803340312306, −27.62777586528275689629745615270, −26.70350264090497031089583410360, −25.47158342427783168874739970089, −25.00072630125429195388039205846, −24.464649527184300595183058077975, −22.76360289048284269578658514867, −21.37353736779857382945914957278, −20.12081296805887060948089948697, −19.879020179822028996301521818722, −18.51225772147183953237508144216, −17.606326087260206913221158659736, −16.30440210708515729351777589371, −15.52377291102313801057798153687, −14.57553747241773298886963469549, −13.19501924852456930737043462350, −12.14297293719791852919140297001, −10.2846145293361730836226667098, −9.26825873269278688535063829474, −8.82402607394587014718235512907, −7.58058147436499770083290067989, −6.23156418000345987936252325659, −4.83621434899337338272958023078, −2.83539976478972037585575443563, −1.51484443410155144022722309051, 1.58521536157015100648544326433, 2.97595769043282299027008347936, 3.82191444676418235301307363781, 6.74090552326993942937391869325, 7.09524159727741762633160942085, 8.73319322003713830029480978292, 9.51771285507114824965945395777, 10.539111933493714426261911668843, 11.62813190163302531662692727860, 13.31230125587287615492354356913, 13.97245572752333753237010579320, 15.35962464906854900042583952694, 16.50211665757945620529902023626, 17.65377888535629064899422870491, 18.794639983814136253282382643633, 19.433755972985574377586467050651, 20.24112277170614309512860431097, 21.49589584367369511280058719929, 22.1939749925339340194306085333, 23.950870196956471444815668419216, 25.10042585343563364745763004771, 25.96907865416708405224266846832, 26.635615145228259997465423377378, 27.182233302997473310802685394493, 29.017294597394116232516472931056

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