Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $0.264 - 0.964i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.142 − 0.989i)3-s + (0.959 − 0.281i)5-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.142 − 0.989i)15-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (−0.841 + 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (0.654 − 0.755i)33-s + ⋯
L(s,χ)  = 1  + (0.142 − 0.989i)3-s + (0.959 − 0.281i)5-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.142 − 0.989i)15-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (−0.841 + 0.540i)21-s + (0.841 − 0.540i)25-s + (−0.415 + 0.909i)27-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (0.654 − 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.264 - 0.964i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.264 - 0.964i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $0.264 - 0.964i$
motivic weight  =  \(0\)
character  :  $\chi_{92} (83, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 92,\ (0:\ ),\ 0.264 - 0.964i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8699320923 - 0.6631196387i$
$L(\frac12,\chi)$  $\approx$  $0.8699320923 - 0.6631196387i$
$L(\chi,1)$  $\approx$  1.033054945 - 0.4513325566i
$L(1,\chi)$  $\approx$  1.033054945 - 0.4513325566i

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

−30.590603920422019803284150590632, −29.3378507837610345072127231880, −28.52009167964194084292726206820, −27.37869859747077600542408831886, −26.39715509289183907115402164487, −25.38238890610317215086020965424, −24.67225826150396045479443123128, −22.64430577181990227915578041375, −22.08440708297201115061082529859, −21.29065221636662376876239679579, −20.01618901279982941645308112182, −18.90062007297593438612889621206, −17.453482373233588609153003344752, −16.58379877447502009505215425230, −15.282235642973892830609609757470, −14.45258856836559178088295450246, −13.17535348484071716382517269260, −11.70932480926545010870221357393, −10.256217758003344275196216268893, −9.56279500228719041893709275400, −8.41362636400138028612654544132, −6.28510851245094473151220477986, −5.45702594261120162886315880564, −3.685813524694320283470891712127, −2.41078521595007270158527949605, 1.33413034198152894220738430795, 2.80587334312072503139050990222, 4.809887759754132055542380554160, 6.54783378214076404684605086372, 7.0954632579116590441277939116, 8.947898974178847682235139484609, 9.826931687241675982210444883761, 11.56266248497447073972693327360, 12.75308002673270881158537386957, 13.67420636677125491874065438063, 14.45751120981544905280718676820, 16.41989369158887725730875839738, 17.33970834354132198264668884515, 18.20454213289824954877051850700, 19.66382137637031813415912478877, 20.18844886461354719433391429129, 21.78416660011564039529639533407, 22.8211134584196502278515789104, 23.99120192990708405560022319380, 24.93614218484064505911991035725, 25.7217595997063384419575631673, 26.787330331188542997956348380711, 28.4972221313524239598069227621, 29.21519580767881848324935665949, 29.957098301265712796850910811922

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