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Properties

Label 2-83-83.82-c12-0-26
Degree $2$
Conductor $83$
Sign $0.870 + 0.492i$
Analytic cond. $75.8614$
Root an. cond. $8.70984$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 102. i·2-s − 81.5·3-s − 6.45e3·4-s + 1.84e4i·5-s + 8.37e3i·6-s + 2.14e5·7-s + 2.41e5i·8-s − 5.24e5·9-s + 1.89e6·10-s − 2.57e6·11-s + 5.25e5·12-s − 4.43e6i·13-s − 2.20e7i·14-s − 1.50e6i·15-s − 1.59e6·16-s − 6.28e6·17-s + ⋯
L(s)  = 1  − 1.60i·2-s − 0.111·3-s − 1.57·4-s + 1.17i·5-s + 0.179i·6-s + 1.82·7-s + 0.922i·8-s − 0.987·9-s + 1.89·10-s − 1.45·11-s + 0.176·12-s − 0.918i·13-s − 2.93i·14-s − 0.131i·15-s − 0.0948·16-s − 0.260·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(83\)
Sign: $0.870 + 0.492i$
Analytic conductor: \(75.8614\)
Root analytic conductor: \(8.70984\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 83,\ (\ :6),\ 0.870 + 0.492i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.472040757\)
\(L(\frac12)\) \(\approx\) \(1.472040757\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 + (-2.84e11 - 1.60e11i)T \)
good2 \( 1 + 102. iT - 4.09e3T^{2} \)
3 \( 1 + 81.5T + 5.31e5T^{2} \)
5 \( 1 - 1.84e4iT - 2.44e8T^{2} \)
7 \( 1 - 2.14e5T + 1.38e10T^{2} \)
11 \( 1 + 2.57e6T + 3.13e12T^{2} \)
13 \( 1 + 4.43e6iT - 2.32e13T^{2} \)
17 \( 1 + 6.28e6T + 5.82e14T^{2} \)
19 \( 1 + 1.47e7iT - 2.21e15T^{2} \)
23 \( 1 + 9.92e7T + 2.19e16T^{2} \)
29 \( 1 - 1.12e9T + 3.53e17T^{2} \)
31 \( 1 + 9.73e8T + 7.87e17T^{2} \)
37 \( 1 - 3.95e9T + 6.58e18T^{2} \)
41 \( 1 - 3.30e9T + 2.25e19T^{2} \)
43 \( 1 + 6.34e9iT - 3.99e19T^{2} \)
47 \( 1 - 1.24e10iT - 1.16e20T^{2} \)
53 \( 1 - 3.84e10iT - 4.91e20T^{2} \)
59 \( 1 + 1.08e10T + 1.77e21T^{2} \)
61 \( 1 + 5.41e10T + 2.65e21T^{2} \)
67 \( 1 - 1.57e11iT - 8.18e21T^{2} \)
71 \( 1 - 1.65e11iT - 1.64e22T^{2} \)
73 \( 1 - 2.06e11iT - 2.29e22T^{2} \)
79 \( 1 - 1.81e11iT - 5.90e22T^{2} \)
89 \( 1 + 3.14e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.21e12iT - 6.93e23T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.33397295039426276378435134176, −10.91043513577571759020734645070, −10.26924924398067496950579612659, −8.552986893508801844676014094003, −7.62000388472297589055518467722, −5.62270555386253642054698471928, −4.47148058388762825625992929997, −2.80599943229311337345687892153, −2.44356567680980766826330985807, −0.905428047374615517730838242485, 0.43018091254042572092582291189, 2.02976892557788699796682460706, 4.67838380978328435687954604612, 4.99352941689634097010894513732, 6.07804871467606602411429304006, 7.81577107442122495780254347475, 8.201972443191723652299017895753, 9.070326610953291405746304675137, 10.94549675929795941084301736297, 12.02297817211145464737682155317

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