↑

Properties

Label 1-131-131.13-r0-0-0
Degree $1$
Conductor $131$
Sign $-0.128 + 0.991i$
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  + (0.644 + 0.764i)2-s + (0.981 − 0.192i)3-s + (−0.168 + 0.985i)4-s + (−0.681 + 0.732i)5-s + (0.779 + 0.626i)6-s + (−0.262 + 0.964i)7-s + (−0.861 + 0.506i)8-s + (0.926 − 0.377i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 − 0.999i)11-s + (0.0241 + 0.999i)12-s + (−0.998 + 0.0483i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (−0.943 − 0.331i)16-s + (0.958 − 0.285i)17-s + ⋯
L(s)  = 1  + (0.644 + 0.764i)2-s + (0.981 − 0.192i)3-s + (−0.168 + 0.985i)4-s + (−0.681 + 0.732i)5-s + (0.779 + 0.626i)6-s + (−0.262 + 0.964i)7-s + (−0.861 + 0.506i)8-s + (0.926 − 0.377i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 − 0.999i)11-s + (0.0241 + 0.999i)12-s + (−0.998 + 0.0483i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (−0.943 − 0.331i)16-s + (0.958 − 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ -0.128 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114382430 + 1.268520651i\)
\(L(\frac12)\) \(\approx\) \(1.114382430 + 1.268520651i\)
\(L(1)\) \(\approx\) \(1.334090029 + 0.8722100714i\)
\(L(1)\) \(\approx\) \(1.334090029 + 0.8722100714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.644 + 0.764i)T \)
3 \( 1 + (0.981 - 0.192i)T \)
5 \( 1 + (-0.681 + 0.732i)T \)
7 \( 1 + (-0.262 + 0.964i)T \)
11 \( 1 + (0.0241 - 0.999i)T \)
13 \( 1 + (-0.998 + 0.0483i)T \)
17 \( 1 + (0.958 - 0.285i)T \)
19 \( 1 + (0.568 - 0.822i)T \)
23 \( 1 + (0.399 + 0.916i)T \)
29 \( 1 + (0.926 + 0.377i)T \)
31 \( 1 + (0.995 + 0.0965i)T \)
37 \( 1 + (-0.443 - 0.896i)T \)
41 \( 1 + (-0.943 + 0.331i)T \)
43 \( 1 + (0.485 + 0.873i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.607 + 0.794i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.485 - 0.873i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (-0.989 - 0.144i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.836 + 0.548i)T \)
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

−28.50184958376319002670067075857, −27.37562356616965832447879556676, −26.752911941251737455304786092506, −25.25052794991304220383867626601, −24.305761898851305744260581637979, −23.31484111703120379688796774145, −22.4218519689561032877229508094, −20.88817464153176666972397196556, −20.47066618516499930937837216424, −19.62303947728959988411452243568, −18.88741657317900603505782654965, −17.062981348146811142138175377808, −15.77225759860918430891468371574, −14.76448642578888543374115756083, −13.891212260442794299871390275662, −12.713922779606049569646291402829, −12.06538445829058801045775560266, −10.254945644787254320378144237240, −9.73177744696829434715575918427, −8.20189454214333914043927527271, −7.04446495225788025015377660406, −4.89197092994855833611480784219, −4.14044693536910559955181737748, −3.002605869959272713036645822960, −1.36571042766411672911064178895, 2.81651317789491124129927265474, 3.3164415233905337616182702904, 4.99690689509801752984176570182, 6.497485597355678293354563996170, 7.5221156823958694187432282832, 8.42674613653018590213613689154, 9.59338068485078108074966985323, 11.57795101364721980267024317947, 12.46356655885351005937646303215, 13.76599873741674601674822240087, 14.562390457936150363534222463975, 15.436828682641522941494226173803, 16.15263379680648723201810992745, 17.84879276409261982416401004465, 18.92384007899392094041203874069, 19.66311597486972445651048599501, 21.31243951384920074835155809934, 21.87923181996876182531745117830, 23.05673008212308821065491402877, 24.18264916976556561061180793601, 24.893113078139491479051276096416, 25.86210562345829530104976798367, 26.69686652731533082590693422640, 27.44628461980181488088920667971, 29.406725751049434391709049300417

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