Properties

Degree 1
Conductor 97
Sign $0.364 + 0.931i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.5 − 0.866i)6-s + (0.258 + 0.965i)7-s i·8-s + (0.5 + 0.866i)9-s + (0.965 + 0.258i)10-s + (−0.866 + 0.5i)11-s + i·12-s + (−0.965 + 0.258i)13-s + (0.258 − 0.965i)14-s + (−0.965 − 0.258i)15-s + (−0.5 + 0.866i)16-s + (0.258 − 0.965i)17-s + ⋯
L(s,χ)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.5 − 0.866i)6-s + (0.258 + 0.965i)7-s i·8-s + (0.5 + 0.866i)9-s + (0.965 + 0.258i)10-s + (−0.866 + 0.5i)11-s + i·12-s + (−0.965 + 0.258i)13-s + (0.258 − 0.965i)14-s + (−0.965 − 0.258i)15-s + (−0.5 + 0.866i)16-s + (0.258 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.364 + 0.931i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.364 + 0.931i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.364 + 0.931i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (9, \cdot )$
Sato-Tate  :  $\mu(24)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 97,\ (0:\ ),\ 0.364 + 0.931i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5870724261 + 0.4008102783i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5870724261 + 0.4008102783i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7518378558 + 0.1952039577i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7518378558 + 0.1952039577i\)

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

−29.87125637160231565857526278875, −28.79895009267848391891698344905, −27.47374746754467811459528983261, −26.53996018353257918552178863975, −26.14646232013108929919561554079, −24.44475729275006898235571377772, −24.13992226716038895377631452023, −23.11955148759518415750584136082, −20.94787286223695441689622295839, −19.929631026225716638901842127386, −19.42564179941384117349653168483, −18.294887844485647209954693890735, −17.11872042229762180459695882080, −15.919845017926893300658330422905, −14.95112513730199115010297446152, −13.896552908900252679968440005678, −12.47427548224523078153357668760, −10.96892435183284205192130529348, −9.77998813432981939101298864741, −8.19437894708431080942700368799, −7.86852368978982587327726965136, −6.69503898844056358333871795072, −4.66598115123994480215086212627, −2.86101596800894527643996685194, −0.90989181190766788227179089878, 2.27661993039650605292184179425, 3.25391627189134863127715247514, 4.82941876990594145233468841551, 7.38924108223940144441532130063, 8.03496273201411531634751657817, 9.27527563411900546187027797158, 10.21963508295659399629167520532, 11.61461172857448182042213035135, 12.48253182219319574809123293505, 14.29157725684114030503327391278, 15.60378503420118029886917255897, 16.01864240452139489199149944938, 17.84433671133465918356211699474, 18.877078292607231169414006972171, 19.5985086407991737174624630848, 20.66584017859548349085729401645, 21.50050194809826951122455335, 22.68402195093205235300755084820, 24.47054777577433236069452923187, 25.34584582378261306625968812934, 26.45338422625415083604304267180, 27.16628707498293402611515359669, 27.93163998944971131001562283463, 29.07711797727111015500274103638, 30.46881350485254556321458926242

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