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Properties

Label 1-97-97.9-r0-0-0
Degree $1$
Conductor $97$
Sign $0.364 + 0.931i$
Analytic cond. $0.450466$
Root an. cond. $0.450466$
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  + (−0.866 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ 0.364 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5870724261 + 0.4008102783i\)
\(L(\frac12)\) \(\approx\) \(0.5870724261 + 0.4008102783i\)
\(L(1)\) \(\approx\) \(0.7518378558 + 0.1952039577i\)
\(L(1)\) \(\approx\) \(0.7518378558 + 0.1952039577i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.965 + 0.258i)T \)
17 \( 1 + (0.258 - 0.965i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.258 + 0.965i)T \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.965 - 0.258i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.965 + 0.258i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 - iT \)
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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

−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