Properties

Degree 1
Conductor 31
Sign $0.544 - 0.838i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 − 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s − 6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.309 − 0.951i)11-s + (0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s,χ)  = 1  + (−0.809 − 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s − 6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.309 − 0.951i)11-s + (0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.544 - 0.838i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.544 - 0.838i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.544 - 0.838i$
motivic weight  =  \(0\)
character  :  $\chi_{31} (23, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 31,\ (1:\ ),\ 0.544 - 0.838i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.263485602 - 0.6861757113i$
$L(\frac12,\chi)$  $\approx$  $1.263485602 - 0.6861757113i$
$L(\chi,1)$  $\approx$  1.053460717 - 0.4046206362i
$L(1,\chi)$  $\approx$  1.053460717 - 0.4046206362i

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

−36.34586104068203256795808869184, −36.18254404947046433419816951247, −33.7627320533376208152470772283, −33.36540498809011424117486673539, −32.26341591888114120131762511493, −30.540233406937276302805905379717, −29.03603797667819291572472798408, −27.727513912860947960025559247302, −26.46002958084717865813369353190, −25.72776733429796362458188710719, −24.65977506247253393639896099802, −23.01611571659203062093857124786, −20.98315621033433075010335712111, −20.24662057477490309204052670546, −18.582300535998321819840690501829, −17.26676079122562009841168679365, −16.05896232245202353545686819417, −14.54112386648540209978080324748, −13.632252253007252212759739530260, −10.62685027631303779612388912885, −9.743978946147242402454810485946, −8.382017691832167994469099926492, −6.78587563857315644526855742999, −4.70537739853280268441677070753, −1.9575110911927671996819340224, 1.63373215918685350596076640681, 3.02314007770844106983583767971, 6.21471622633550805039157433271, 8.2665872422878573757429102791, 9.06655662913371574782475996910, 10.76140316235983546624840629694, 12.5661378758048446697706336889, 13.65561019591491120204956644703, 15.51449148904393402888051974522, 17.51633508992540963445166005648, 18.39551380196769157359724300330, 19.49171642349767974418796295770, 21.01510923350339695046907606948, 21.70233834070759406714664924548, 24.241850570423611535179623319689, 25.41840823263600356123861538673, 26.07897950654395946136932602383, 27.73608994586007887660688993422, 29.01951652168037335796314888442, 30.005394132694299012226287795271, 31.060319053779928034103084912540, 32.43310552336537297609135857137, 34.33112090214537823899856566049, 35.33161020921246885381153605777, 36.687546405675858072788100419069

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