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Properties

Label 1-97-97.58-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.980 - 0.194i$
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  + (−0.608 − 0.793i)2-s + (−0.130 + 0.991i)3-s + (−0.258 + 0.965i)4-s + (−0.997 + 0.0654i)5-s + (0.866 − 0.5i)6-s + (0.442 + 0.896i)7-s + (0.923 − 0.382i)8-s + (−0.965 − 0.258i)9-s + (0.659 + 0.751i)10-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)12-s + (0.0654 + 0.997i)13-s + (0.442 − 0.896i)14-s + (0.0654 − 0.997i)15-s + (−0.866 − 0.5i)16-s + (−0.896 − 0.442i)17-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.130 + 0.991i)3-s + (−0.258 + 0.965i)4-s + (−0.997 + 0.0654i)5-s + (0.866 − 0.5i)6-s + (0.442 + 0.896i)7-s + (0.923 − 0.382i)8-s + (−0.965 − 0.258i)9-s + (0.659 + 0.751i)10-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)12-s + (0.0654 + 0.997i)13-s + (0.442 − 0.896i)14-s + (0.0654 − 0.997i)15-s + (−0.866 − 0.5i)16-s + (−0.896 − 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.980 - 0.194i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.980 - 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01944097427 + 0.1978996545i\)
\(L(\frac12)\) \(\approx\) \(0.01944097427 + 0.1978996545i\)
\(L(1)\) \(\approx\) \(0.5125350931 + 0.1099540687i\)
\(L(1)\) \(\approx\) \(0.5125350931 + 0.1099540687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.608 - 0.793i)T \)
3 \( 1 + (-0.130 + 0.991i)T \)
5 \( 1 + (-0.997 + 0.0654i)T \)
7 \( 1 + (0.442 + 0.896i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (0.0654 + 0.997i)T \)
17 \( 1 + (-0.896 - 0.442i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (-0.321 - 0.946i)T \)
29 \( 1 + (-0.659 + 0.751i)T \)
31 \( 1 + (-0.991 - 0.130i)T \)
37 \( 1 + (-0.946 - 0.321i)T \)
41 \( 1 + (-0.751 - 0.659i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.793 - 0.608i)T \)
59 \( 1 + (0.321 - 0.946i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (-0.751 + 0.659i)T \)
73 \( 1 + (-0.258 - 0.965i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.442 + 0.896i)T \)
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

−29.34871828453978576406699490485, −27.88586980144593559071171503492, −27.2723336390191564714317494446, −26.17413013236368988825331826987, −24.91673893495723829925323285334, −24.136958021217367058513534618075, −23.3942548315715317716743555342, −22.531094923575145958291108615741, −20.16680852809051029612110121246, −19.59638685032432628442081672229, −18.57516638575118946158596260398, −17.40700060220493683992341343704, −16.71813496845973575671311078194, −15.31312789352304501283119235595, −14.23335103850980723842162217193, −13.11892652966158188800579242136, −11.552447278496076127756518254973, −10.62038186543759833690992942576, −8.66022009493243882387032578454, −7.858147460338105461835347595220, −6.977881525332766902196996057693, −5.699722818861764551870934753326, −3.94338914200933510878046115978, −1.40327444431266427808833122441, −0.11708075389643105107298665305, 2.27308333556590212440179083438, 3.87512964883210565695693400782, 4.72841678993408657389295579165, 6.98353442524505070545942350958, 8.73991886615209780489772869020, 9.12988590411199572630257009973, 10.80274934932098680437202024428, 11.54230849698935785363281667610, 12.381112925026424670897492738, 14.40603388473113439660208425715, 15.51667052273256037649775952205, 16.51208479371843776569807371665, 17.67159904609281719434033470592, 18.87889046851405834693763745315, 19.9313640650682418567516159635, 20.76132467941228302904436715855, 21.98869290294290111654167223385, 22.4949425329935589399577488061, 24.088360893577843566891343004750, 25.65848527585400247352645951965, 26.57707174082698336426425101232, 27.57988363398567171087273240179, 28.016581434888282423457851666976, 28.96717944347403645687333861021, 30.637228411278953292101177908158

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