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Properties

Label 1-97-97.46-r1-0-0
Degree $1$
Conductor $97$
Sign $0.880 + 0.474i$
Analytic cond. $10.4240$
Root an. cond. $10.4240$
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  + (−0.382 + 0.923i)2-s + (0.923 − 0.382i)3-s + (−0.707 − 0.707i)4-s + (0.555 + 0.831i)5-s + i·6-s + (0.555 − 0.831i)7-s + (0.923 − 0.382i)8-s + (0.707 − 0.707i)9-s + (−0.980 + 0.195i)10-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)12-s + (0.831 − 0.555i)13-s + (0.555 + 0.831i)14-s + (0.831 + 0.555i)15-s + i·16-s + (0.831 − 0.555i)17-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (0.923 − 0.382i)3-s + (−0.707 − 0.707i)4-s + (0.555 + 0.831i)5-s + i·6-s + (0.555 − 0.831i)7-s + (0.923 − 0.382i)8-s + (0.707 − 0.707i)9-s + (−0.980 + 0.195i)10-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)12-s + (0.831 − 0.555i)13-s + (0.555 + 0.831i)14-s + (0.831 + 0.555i)15-s + i·16-s + (0.831 − 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.880 + 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.064089493 + 0.5210123696i\)
\(L(\frac12)\) \(\approx\) \(2.064089493 + 0.5210123696i\)
\(L(1)\) \(\approx\) \(1.329411519 + 0.3332615065i\)
\(L(1)\) \(\approx\) \(1.329411519 + 0.3332615065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.382 + 0.923i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (0.555 + 0.831i)T \)
7 \( 1 + (0.555 - 0.831i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
17 \( 1 + (0.831 - 0.555i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (0.980 + 0.195i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (-0.980 + 0.195i)T \)
61 \( 1 + T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.555 - 0.831i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
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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

−29.72124546354202760818760383146, −28.508247168380182401297405195209, −27.89288117327684627661546852727, −26.77150237647038651409879606257, −25.692749505361565375087600242768, −24.925251728352450503616700388747, −23.46897093762455901129432888091, −21.62681799242101599695021430914, −21.13432008434930700869806400704, −20.60153515465695146988604403058, −19.1051107480986606501198849000, −18.46615420671630709655892939662, −17.016650646495784533368686517034, −15.88253278184059061811755927585, −14.35589852899481842933653435896, −13.30374563015613735916458360320, −12.37415486114464346691060806970, −10.84491372448922082877788660339, −9.72644251576486190878431365038, −8.61426220760212480601149201373, −8.145049219433994888844633393460, −5.45503245732759461294779779388, −4.160958718604469878191472712410, −2.623613412437469923272215168366, −1.48206288082311413993840471171, 1.23108756438341136995186876308, 3.05741953518980548275982908129, 4.89613886097889774196261681819, 6.584192539169943405468303027953, 7.4647777737270602614975070776, 8.43641741713374168786125750395, 9.871012548194740730476582341931, 10.7395177665996568126889915573, 13.159564079886469301384741959792, 13.84512211611029272075019734603, 14.795159206575725386898785251954, 15.70066209771268611250993854323, 17.37265629799103574769082902757, 18.14155613916879927602680591312, 18.97577183764338969335772314312, 20.30120039528481235338212708333, 21.383409861697455153450898039240, 23.14475445103258748906238749775, 23.64639705084157606175388005318, 25.15547628195202335539834354371, 25.6291521997432763777870609504, 26.572654838927487962674635656020, 27.33653179312267139247932407180, 28.913543993277015713309399635401, 30.1719020962734062986603416452

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