Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s − 5-s + 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s + 25-s − 27-s + 29-s − 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s − 57-s + ⋯
L(s,χ)  = 1  − 3-s − 5-s + 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s + 25-s − 27-s + 29-s − 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s − 57-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{92} (91, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 92,\ (0:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7475038909\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7475038909\)
\(L(\chi,1)\)  \(\approx\)  \(0.8071106504\)
\(L(1,\chi)\)  \(\approx\)  \(0.8071106504\)

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

−30.56546016175685802216170537662, −29.242823378487636721827334521395, −28.006455842521963452729555056551, −27.50463251533811286246445992468, −26.54607826599937009595547350927, −24.70249611236044990043694110744, −23.99145708958307193176741489049, −22.994177604884986285126178705588, −22.12474197925929969063632827160, −20.84537689176568927077148588205, −19.67494936955169080576522087599, −18.35144023819853553867980194098, −17.53328555413036545501775757087, −16.267668943364381410263844552361, −15.41751803819302293367611100153, −14.03440772808988137627006644797, −12.42016655278868888457002496931, −11.43829100805393573820535556358, −10.880488562437947285332371309965, −8.99041044322388354298083545149, −7.63725952967221761670663025449, −6.426680777461463525192446696654, −4.90215948742937868411286451809, −3.857973350417766331784296841016, −1.28645787889364417182140347191, 1.28645787889364417182140347191, 3.857973350417766331784296841016, 4.90215948742937868411286451809, 6.426680777461463525192446696654, 7.63725952967221761670663025449, 8.99041044322388354298083545149, 10.880488562437947285332371309965, 11.43829100805393573820535556358, 12.42016655278868888457002496931, 14.03440772808988137627006644797, 15.41751803819302293367611100153, 16.267668943364381410263844552361, 17.53328555413036545501775757087, 18.35144023819853553867980194098, 19.67494936955169080576522087599, 20.84537689176568927077148588205, 22.12474197925929969063632827160, 22.994177604884986285126178705588, 23.99145708958307193176741489049, 24.70249611236044990043694110744, 26.54607826599937009595547350927, 27.50463251533811286246445992468, 28.006455842521963452729555056551, 29.242823378487636721827334521395, 30.56546016175685802216170537662

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