Properties

Degree 1
Conductor 131
Sign $-0.128 + 0.991i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $-0.128 + 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (13, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ -0.128 + 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.114382430 + 1.268520651i$
$L(\frac12,\chi)$  $\approx$  $1.114382430 + 1.268520651i$
$L(\chi,1)$  $\approx$  1.334090029 + 0.8722100714i
$L(1,\chi)$  $\approx$  1.334090029 + 0.8722100714i

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

−28.50184958376319002670067075857, −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