Properties

Degree 1
Conductor 97
Sign $-0.393 + 0.919i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (0.923 + 0.382i)5-s − 6-s + (0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s i·9-s + (0.382 + 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.923 − 0.382i)13-s + (0.923 + 0.382i)14-s + (−0.923 + 0.382i)15-s − 16-s + (−0.923 − 0.382i)17-s + ⋯
L(s,χ)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (0.923 + 0.382i)5-s − 6-s + (0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s i·9-s + (0.382 + 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.923 − 0.382i)13-s + (0.923 + 0.382i)14-s + (−0.923 + 0.382i)15-s − 16-s + (−0.923 − 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.393 + 0.919i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.393 + 0.919i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $-0.393 + 0.919i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (8, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (0:\ ),\ -0.393 + 0.919i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7195034467 + 1.090196433i$
$L(\frac12,\chi)$  $\approx$  $0.7195034467 + 1.090196433i$
$L(\chi,1)$  $\approx$  1.018368144 + 0.8371583131i
$L(1,\chi)$  $\approx$  1.018368144 + 0.8371583131i

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

−29.564796266244381009029956536627, −28.980861237292382987403711406802, −28.25127910528354103067603415378, −26.98016684680569563954355110944, −25.0631042817907373943988270597, −24.21579549095195024553292587952, −23.72743768575642131837322576264, −22.01106167052246786614851879134, −21.71814322319928272393461316171, −20.456318616819286740330109896104, −19.147085488686319935081795904244, −18.09128135776569827477593326422, −17.28300416571397953599215982583, −15.68395885387361440964850244528, −14.09427568294314108985710501487, −13.40579377537937880255371577283, −12.280228323891796535920251178687, −11.33962213424888842159407576014, −10.27399631768084377518061281693, −8.71562336282065583657317511106, −6.86879751724370292912608365322, −5.432966084220273174244268696726, −4.97545871934902319279881368531, −2.53071296480721811866723847296, −1.41679223254449677072988432355, 2.614512229051286648511670053163, 4.5378882672659743431410016728, 5.20914030715717297820385352924, 6.49262156995221291145048925454, 7.73119254724387419455558449843, 9.51780530270899800744507429658, 10.68287637843254956573041086150, 11.90777282512552211333410301380, 13.2288463446882322166052856871, 14.47003046637500504491345148376, 15.21332886593742392135639936240, 16.48468284714363968708170510861, 17.66320752700602947486089483433, 17.879509874591297242872622886690, 20.5859880596814999919045202855, 21.101808043423506605071212955009, 22.32015302673595627292643128891, 22.83104368270275699712865322250, 24.16658726965734530026954579927, 25.00862908132174696022558976000, 26.53571439731544595717508101066, 26.80353105001990405966987786478, 28.46114664147683213436409187617, 29.476377316893449692228653046388, 30.4888900310269153691159329209

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