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Properties

Label 1-31-31.23-r1-0-0
Degree $1$
Conductor $31$
Sign $0.544 - 0.838i$
Analytic cond. $3.33141$
Root an. cond. $3.33141$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.544 - 0.838i$
Analytic conductor: \(3.33141\)
Root analytic conductor: \(3.33141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ 0.544 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263485602 - 0.6861757113i\)
\(L(\frac12)\) \(\approx\) \(1.263485602 - 0.6861757113i\)
\(L(1)\) \(\approx\) \(1.053460717 - 0.4046206362i\)
\(L(1)\) \(\approx\) \(1.053460717 - 0.4046206362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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