Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $-0.293 + 0.956i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.293 + 0.956i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.293 + 0.956i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $-0.293 + 0.956i$
motivic weight  =  \(0\)
character  :  $\chi_{92} (59, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 92,\ (1:\ ),\ -0.293 + 0.956i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.03205446198 + 0.04334947363i$
$L(\frac12,\chi)$  $\approx$  $0.03205446198 + 0.04334947363i$
$L(\chi,1)$  $\approx$  0.5672425983 - 0.2168791443i
$L(1,\chi)$  $\approx$  0.5672425983 - 0.2168791443i

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

−29.608544481102853695188230230434, −28.63279954212661400458995045577, −27.700566412867025369837643964358, −26.49116799978401800905189225479, −26.13502610550270559477679405449, −24.43687397181695660952779532195, −23.134713518296406579482894272737, −22.33206895520508819745119385713, −21.617181898221299098329583017704, −20.206021267896913765246048634074, −19.03873654024660254929937801529, −18.0753852777969686956855441839, −16.386473589331503213636175939929, −15.85992877714988686009452716042, −14.8093490553990374700818800893, −13.414593160753678873666088244941, −11.62612769056759329321838670279, −11.1338562455458090370177393947, −9.690469839225809713611447933875, −8.627390547527390057043442842458, −6.76035741145408786622345816478, −5.72846130675755650232921954377, −3.97608070231404472150270462634, −3.0234035064113433884954816974, −0.02666756667046684328547814893, 1.47065358511391874125696918336, 3.586343047280773912186090718401, 5.16778944999664416858687989329, 6.66150727616846477276422084060, 7.6460874684276345218335422520, 8.955719779541627995346621707078, 10.575644936459644887873465641881, 11.91118730720346920180904774764, 12.812825096627019922328254096984, 13.609214121069361117917912825517, 15.43762529017711407902316535888, 16.47417208806888063402178777919, 17.489360596258954437873666941311, 18.631339304605737655002903322989, 19.90269483475767145419761186946, 20.336342152685759454613026465808, 22.41130366954420493217085035257, 23.093763789451482768103706645371, 24.000985795523149731311371438480, 25.03762262745029541480491676139, 26.04575126863347457964098115264, 27.540502325081611666163510594766, 28.4932677616779296306365880177, 29.204784730390528400765976019166, 30.53273715745721181760654698313

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