Properties

Degree 1
Conductor 67
Sign $0.283 + 0.959i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.786 + 0.618i)2-s + (−0.654 + 0.755i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.0475 − 0.998i)6-s + (0.928 + 0.371i)7-s + (0.415 + 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.327 + 0.945i)10-s + (0.0475 + 0.998i)11-s + (0.580 + 0.814i)12-s + (−0.995 + 0.0950i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (0.235 + 0.971i)17-s + ⋯
L(s,χ)  = 1  + (−0.786 + 0.618i)2-s + (−0.654 + 0.755i)3-s + (0.235 − 0.971i)4-s + (0.841 − 0.540i)5-s + (0.0475 − 0.998i)6-s + (0.928 + 0.371i)7-s + (0.415 + 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.327 + 0.945i)10-s + (0.0475 + 0.998i)11-s + (0.580 + 0.814i)12-s + (−0.995 + 0.0950i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (−0.888 − 0.458i)16-s + (0.235 + 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.283 + 0.959i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.283 + 0.959i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $0.283 + 0.959i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (56, \cdot )$
Sato-Tate  :  $\mu(33)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 67,\ (0:\ ),\ 0.283 + 0.959i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5116210080 + 0.3824134433i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5116210080 + 0.3824134433i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6508603162 + 0.3115735324i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6508603162 + 0.3115735324i\)

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

−31.31663363627132706445819317309, −30.19430096766526233820871946449, −29.498275236458091783266251652463, −28.877382097555655682250605230508, −27.377980862003480697580033909321, −26.658370794016340068906460829754, −25.07626100062561817942368830801, −24.39592474753478529480308692338, −22.655424212081156817979167662685, −21.71527705957596499623672500136, −20.571047551921477889068385682429, −19.07215492241585650509054547071, −18.25571070878207072397800913490, −17.37632043420776959462020239647, −16.53784655422826880189149828929, −14.25706445442116852613897594257, −13.17828453499795267148072134784, −11.670530138065214141310213461910, −10.9439128483701088388861991459, −9.6549646307402747732252962982, −7.934430775507692782876974475750, −6.92510580785494756973639825051, −5.24873633831191262004010383208, −2.77644808941050572530356814260, −1.30573367910127874042384673796, 1.73393371002523266968049104033, 4.87707164385907087732548072677, 5.5282736102232391813432103485, 7.18457258664405498497523274124, 8.917658293826800817898006415883, 9.75718964277253090260535305797, 10.916037915155044189430855362213, 12.36852962459091072852452104866, 14.452918374746916554439865098181, 15.24039651595282928092580347824, 16.67998524542815225915458986583, 17.42082952620156083898981271476, 18.13869202660954032422179554343, 20.02028162619582332396245195017, 21.05999613884446391166397862923, 22.19725699254338371440428332518, 23.685334839911422874900522137943, 24.60724951176182595125327626057, 25.72330501456284272233296006818, 26.8836862098245404582012592169, 27.9309997256781868371016524387, 28.50199533061421009425901343546, 29.58008925460400875689777280194, 31.46088487296462683791568544089, 32.888805194309940621231291840990

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