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Properties

Label 1-83-83.3-r0-0-0
Degree $1$
Conductor $83$
Sign $0.756 - 0.654i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
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  + (0.720 − 0.693i)2-s + (0.190 + 0.981i)3-s + (0.0383 − 0.999i)4-s + (−0.264 − 0.964i)5-s + (0.817 + 0.575i)6-s + (0.988 + 0.152i)7-s + (−0.665 − 0.746i)8-s + (−0.927 + 0.373i)9-s + (−0.859 − 0.511i)10-s + (0.896 + 0.443i)11-s + (0.988 − 0.152i)12-s + (−0.771 − 0.636i)13-s + (0.817 − 0.575i)14-s + (0.896 − 0.443i)15-s + (−0.997 − 0.0765i)16-s + (0.477 + 0.878i)17-s + ⋯
L(s)  = 1  + (0.720 − 0.693i)2-s + (0.190 + 0.981i)3-s + (0.0383 − 0.999i)4-s + (−0.264 − 0.964i)5-s + (0.817 + 0.575i)6-s + (0.988 + 0.152i)7-s + (−0.665 − 0.746i)8-s + (−0.927 + 0.373i)9-s + (−0.859 − 0.511i)10-s + (0.896 + 0.443i)11-s + (0.988 − 0.152i)12-s + (−0.771 − 0.636i)13-s + (0.817 − 0.575i)14-s + (0.896 − 0.443i)15-s + (−0.997 − 0.0765i)16-s + (0.477 + 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.350239114 - 0.5029054126i\)
\(L(\frac12)\) \(\approx\) \(1.350239114 - 0.5029054126i\)
\(L(1)\) \(\approx\) \(1.423131047 - 0.3796353091i\)
\(L(1)\) \(\approx\) \(1.423131047 - 0.3796353091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (0.720 - 0.693i)T \)
3 \( 1 + (0.190 + 0.981i)T \)
5 \( 1 + (-0.264 - 0.964i)T \)
7 \( 1 + (0.988 + 0.152i)T \)
11 \( 1 + (0.896 + 0.443i)T \)
13 \( 1 + (-0.771 - 0.636i)T \)
17 \( 1 + (0.477 + 0.878i)T \)
19 \( 1 + (-0.114 + 0.993i)T \)
23 \( 1 + (-0.409 - 0.912i)T \)
29 \( 1 + (-0.973 - 0.227i)T \)
31 \( 1 + (-0.665 + 0.746i)T \)
37 \( 1 + (-0.927 - 0.373i)T \)
41 \( 1 + (0.720 + 0.693i)T \)
43 \( 1 + (-0.543 + 0.839i)T \)
47 \( 1 + (0.338 + 0.941i)T \)
53 \( 1 + (0.338 - 0.941i)T \)
59 \( 1 + (0.606 - 0.795i)T \)
61 \( 1 + (0.953 - 0.301i)T \)
67 \( 1 + (-0.997 - 0.0765i)T \)
71 \( 1 + (0.988 - 0.152i)T \)
73 \( 1 + (-0.859 - 0.511i)T \)
79 \( 1 + (0.0383 - 0.999i)T \)
89 \( 1 + (0.817 + 0.575i)T \)
97 \( 1 + (0.817 - 0.575i)T \)
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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

−31.08237358632534837930841279364, −29.93867308917200524146281404501, −29.71565062945348366910613095565, −27.5110421032501253461253541646, −26.465446065785720391812380841474, −25.4844859991409641546831957654, −24.32264433408282859778910158771, −23.80758568444707180321017221621, −22.58974262745288635970586100254, −21.66907537936183708262188169127, −20.17557349829009887443062486751, −18.89433931491522744916823728385, −17.79095715429607485085020273883, −16.88163567060693469675737836556, −15.12546832851504047454786151446, −14.28463060042931535937176001998, −13.64519636959626560006531367460, −11.89776971321167994872408212981, −11.40556957711439779355898735150, −8.927545496649826497651461916995, −7.472592550887283017316056144996, −7.00256562942041547873852387152, −5.525400372206082410521670033388, −3.7767915679979404875924863998, −2.28320476474988652166762309155, 1.79824851956846894716883789533, 3.74375536429294684505323520029, 4.67708726235557788191896608563, 5.65737763772288091040608606649, 8.138378574223536106106820441156, 9.39006929620544374430879194282, 10.51948221934827435181621271544, 11.78573744105263097598836097042, 12.66064084826832960361561972312, 14.46086289470583400851455261099, 14.859444585783648205541691600193, 16.31510976331082994411197686093, 17.49777485723318100704782615553, 19.375324751360762775523782137377, 20.31485474153895776063243581234, 20.951756975276791729444690447593, 21.96578941049679557559075198814, 23.00380997977086349746106248951, 24.29554889252870367134039936660, 25.113894834478371935826717420670, 27.050333920431747100139112714152, 27.79056978004054889290009925221, 28.35851575861630298016960874934, 29.8664928037283905457044024384, 30.96717773029685844191163377138

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