Properties

Degree 1
Conductor 83
Sign $0.756 + 0.654i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.756 + 0.654i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (28, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.756 + 0.654i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.350239114 + 0.5029054126i$
$L(\frac12,\chi)$  $\approx$  $1.350239114 + 0.5029054126i$
$L(\chi,1)$  $\approx$  1.423131047 + 0.3796353091i
$L(1,\chi)$  $\approx$  1.423131047 + 0.3796353091i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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