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Properties

Label 1-31-31.22-r1-0-0
Degree $1$
Conductor $31$
Sign $0.987 + 0.159i$
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 + (−0.913 − 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (0.978 + 0.207i)11-s + (0.104 − 0.994i)12-s + (0.104 + 0.994i)13-s + (−0.978 + 0.207i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.978 − 0.207i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.913 − 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (0.978 + 0.207i)11-s + (0.104 − 0.994i)12-s + (0.104 + 0.994i)13-s + (−0.978 + 0.207i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.978 − 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(3.33141\)
Root analytic conductor: \(3.33141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7022495243 + 0.05637409470i\)
\(L(\frac12)\) \(\approx\) \(0.7022495243 + 0.05637409470i\)
\(L(1)\) \(\approx\) \(0.6202467172 - 0.05985226335i\)
\(L(1)\) \(\approx\) \(0.6202467172 - 0.05985226335i\)

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

−36.2522960507084145325954297370, −34.826111336646171946774448193861, −34.593170698638995805689464858010, −32.89363300430623264508359146105, −32.16647466958680711173927161117, −30.02946690346975909752496985490, −28.3064304639656497189162314094, −27.90808326625986959516955632933, −26.93346691293924097908014392486, −25.00399233464050259355059118338, −24.13970476949148202062787359738, −22.87607989284162025649510252814, −21.17091458259031212326590134544, −19.69012179489032358081079039430, −18.08877903781105312218405693279, −17.04817387003324079448239619211, −15.960729740727348081646457283196, −14.84761695620097159643897551566, −12.25255051944786846348513052556, −11.064355719876936311520253835049, −9.37512357983081562739443310323, −8.06276597845191380458038721800, −6.058605118724657326368452102786, −4.785421228510313257053987873572, −0.86734042277587321242882642972, 1.42724419321017576941360630715, 3.98399601539889279516376515714, 6.72642466416705714583734708087, 7.79530942240813706616693139156, 10.05733132093505013976163558488, 11.293537220783265144132886521030, 12.0565183581053145854500972254, 14.14829751939124552274094388930, 16.32940976421178088738407126061, 17.39808738251428242890294361764, 18.55459192220367026910783856876, 19.56737128813688947762187121293, 21.28251709821787186217446922844, 22.59612313050221347538468062052, 23.81008030009574115476912282889, 25.51918721661136353710050655569, 27.117107115587394421124069905322, 27.63606649473705546323285706009, 29.26481552316304555329390952142, 30.095417344213222757939348948365, 30.961941483346572583639281100692, 33.51268828389573773948219880226, 34.25856992256049194006659899691, 35.47503640768184046308857716618, 36.319238269800547231247325605061

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