Properties

Label 1-31-31.20-r0-0-0
Degree $1$
Conductor $31$
Sign $-0.604 + 0.796i$
Analytic cond. $0.143963$
Root an. cond. $0.143963$
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 + (−0.104 + 0.994i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (0.669 + 0.743i)11-s + (0.913 + 0.406i)12-s + (0.913 − 0.406i)13-s + (0.669 − 0.743i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.669 − 0.743i)17-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.104 + 0.994i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (0.669 + 0.743i)11-s + (0.913 + 0.406i)12-s + (0.913 − 0.406i)13-s + (0.669 − 0.743i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.669 − 0.743i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $-0.604 + 0.796i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ -0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1991632166 + 0.4010864112i\)
\(L(\frac12)\) \(\approx\) \(0.1991632166 + 0.4010864112i\)
\(L(1)\) \(\approx\) \(0.4515293016 + 0.3943007268i\)
\(L(1)\) \(\approx\) \(0.4515293016 + 0.3943007268i\)

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 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.913 - 0.406i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
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

−36.17544753793386369783800007059, −35.4072580441796122382534135979, −34.731546842609679255158578041822, −32.53836983556514297342012883642, −31.10084512061844517431714857394, −30.00360286960329453477082336644, −28.83638488602228454958944700281, −28.08991154244444441648225340610, −26.44181962703953329506332515199, −25.20586496146279754310082231991, −23.99568726583195979097293085612, −22.48393053556971025409800272981, −20.63455765387678887955098675592, −19.478001555937092583680958193664, −18.76370345171913444247532723068, −17.04610307959612534830308096495, −16.21817782138821658404278705961, −13.46073606637374061359398965025, −12.412325838087355522215203576311, −11.26009480707037679870443917191, −9.201544748730044035377593896283, −8.03689453383942110138629714492, −6.4276988293955424104253576982, −3.47167474507840179793151472467, −1.100148492494785447431532988451, 3.4187048448422688456194145716, 5.74284126321691369338654880447, 7.27314917182395729881768357459, 9.18645579116628708017627918036, 10.21279093390690613521337702032, 11.57167656714932540999543295535, 14.32133379179052035490221336546, 15.508707777237651210436166139754, 16.29133423944569065859467695929, 17.889019659470139420285862721115, 19.24276256119616348481805989425, 20.44722826501447489017838011460, 22.50034671831469411927423953910, 23.11179038120913312256880680141, 25.3396040787606141805274766918, 26.06529086873843906755747371673, 27.28292481980241567720003971917, 28.09289757587272299545185740700, 29.52573433486529770880510115484, 31.41962530416648618740826247448, 32.787294138571927053994561742077, 33.60422829030002860036916166493, 34.86373753237590016340876449647, 35.689963066360949574821666417, 37.56117296713016948724792621859

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