Properties

Degree 1
Conductor 31
Sign $0.987 + 0.159i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 − 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.987 + 0.159i$
motivic weight  =  \(0\)
character  :  $\chi_{31} (22, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 31,\ (1:\ ),\ 0.987 + 0.159i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7022495243 + 0.05637409470i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7022495243 + 0.05637409470i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6202467172 - 0.05985226335i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6202467172 - 0.05985226335i\)

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

−36.2522960507084145325954297370, −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