Properties

Degree 1
Conductor 83
Sign $-0.675 + 0.737i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.665 + 0.746i)2-s + (−0.543 − 0.839i)3-s + (−0.114 + 0.993i)4-s + (−0.720 − 0.693i)5-s + (0.264 − 0.964i)6-s + (0.896 + 0.443i)7-s + (−0.817 + 0.575i)8-s + (−0.409 + 0.912i)9-s + (0.0383 − 0.999i)10-s + (0.190 + 0.981i)11-s + (0.896 − 0.443i)12-s + (−0.477 + 0.878i)13-s + (0.264 + 0.964i)14-s + (−0.190 + 0.981i)15-s + (−0.973 − 0.227i)16-s + (−0.997 − 0.0765i)17-s + ⋯
L(s,χ)  = 1  + (0.665 + 0.746i)2-s + (−0.543 − 0.839i)3-s + (−0.114 + 0.993i)4-s + (−0.720 − 0.693i)5-s + (0.264 − 0.964i)6-s + (0.896 + 0.443i)7-s + (−0.817 + 0.575i)8-s + (−0.409 + 0.912i)9-s + (0.0383 − 0.999i)10-s + (0.190 + 0.981i)11-s + (0.896 − 0.443i)12-s + (−0.477 + 0.878i)13-s + (0.264 + 0.964i)14-s + (−0.190 + 0.981i)15-s + (−0.973 − 0.227i)16-s + (−0.997 − 0.0765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.675 + 0.737i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.675 + 0.737i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.675 + 0.737i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (56, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.675 + 0.737i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4742898903 + 1.077873371i$
$L(\frac12,\chi)$  $\approx$  $0.4742898903 + 1.077873371i$
$L(\chi,1)$  $\approx$  0.9225547616 + 0.4422790722i
$L(1,\chi)$  $\approx$  0.9225547616 + 0.4422790722i

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

−30.18430192680673666476861767156, −29.3689029546201783235222480860, −28.01158620441852712123910536222, −27.22295605144344050730253063362, −26.56535438413316052178264830766, −24.3544347178869280507164914407, −23.53434618437283150681624044131, −22.43742971242981973294122425408, −21.82004709539735560806935962905, −20.61451815904537720066684022734, −19.69217234446161883476917288477, −18.34902803545136910600800354630, −17.08667110579407341418718429821, −15.39320024252479690500003319994, −14.90333093208374548241533135611, −13.53805573897517718394492990967, −11.86185509766517424116118551583, −11.02613963804029449372824121290, −10.47598915970516342433006339695, −8.72309391291385331569622192401, −6.75041920378643476648516021554, −5.2071413767296651576237022062, −4.15555165931202445264518946500, −2.94539367723187023724193950786, −0.47662007499814702052013852015, 1.95393013021167537673326980814, 4.36727810183604151622735899642, 5.22409733676094735016528062060, 6.784172864134727800227658917459, 7.76486389222526564732806706538, 8.84318023891409901502624599019, 11.45256944885311779006723661804, 12.11096036018282596106345720046, 13.07081311318845134422742863641, 14.451788117767660134379904618881, 15.52673809839532028662057323670, 16.86452047261511748115275280978, 17.54244439062111879128553254341, 18.86725374201512032863753802421, 20.321947366217439037268725988277, 21.546642335971833138368998065, 22.870996539672918884611993372226, 23.57624282696164269079615137879, 24.59123341778228398626340158517, 25.0117887184132053382102046366, 26.73559590987494191018677004043, 27.8584278613703143247049187508, 28.91046283257540559154633180553, 30.36747413705287147444104447564, 31.09605903845581068537531506160

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