Properties

Degree 1
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.984 + 0.173i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + 35-s + 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + 35-s + 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.984 + 0.173i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.984 + 0.173i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.984 + 0.173i$
motivic weight  =  \(0\)
character  :  $\chi_{36} (23, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 36,\ (0:\ ),\ 0.984 + 0.173i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8222820649 + 0.07194035890i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8222820649 + 0.07194035890i\)
\(L(\chi,1)\)  \(\approx\)  \(1.003837166 + 0.05783774636i\)
\(L(1,\chi)\)  \(\approx\)  \(1.003837166 + 0.05783774636i\)

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

−35.86690603875864830574326672474, −34.48935673580116156526442706543, −33.43701017854223800184568857670, −31.980390955335435978492680125656, −31.27588267842812972985599510952, −29.5081392852163109266857295296, −28.57261072631808318262395139322, −27.43679296486427561565187471753, −25.93193174639440572944941739710, −24.59162420280926346667433058825, −23.870700345723942497068926763896, −21.79998852975647480543378572569, −21.18031806028247547499431076424, −19.586675629076211234362596893133, −18.169039935900764697177121392176, −16.889791040762615134076772847028, −15.58071229483743724297034263903, −13.97502385043088870723119736977, −12.648325869272542811061859051250, −11.25279823282029673209891658310, −9.35390753944101506765847727587, −8.278342933690077090023168122797, −6.06203944896421696376204364589, −4.676075508748655458490821851313, −2.14409489942574708228920090773, 2.41511530935627412534947040506, 4.55075499877039430801681565928, 6.5261700824006092798773189112, 7.881531411731270516392470116776, 10.01002603795048827922562440232, 10.91146184578109010597210076996, 12.85216184693912363313832213407, 14.22256159954633033580274083309, 15.34094671625210298882315145436, 17.29453928386547325791063122901, 18.050841047001815437271326540424, 19.73252620095592696928190534912, 20.96480270317158436691406526239, 22.36886154269478372052879267445, 23.41316604449680327709302166251, 24.94746180270863493885699970595, 26.19713634504464078781560156483, 27.137992170147889144606728655683, 28.70783783763675238999960959202, 30.00072573157095372216919796400, 30.709460580858837860147969780235, 32.410377116432291659288084035111, 33.59985372037602719440421343101, 34.32559475708389148454345625691, 36.00525677501136349756372020648

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