Properties

Degree 1
Conductor 83
Sign $0.582 + 0.813i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.997 + 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (0.477 + 0.878i)5-s + (−0.771 − 0.636i)6-s + (0.817 − 0.575i)7-s + (−0.973 + 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (−0.927 − 0.373i)13-s + (−0.771 + 0.636i)14-s + (−0.264 + 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯
L(s,χ)  = 1  + (−0.997 + 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (0.477 + 0.878i)5-s + (−0.771 − 0.636i)6-s + (0.817 − 0.575i)7-s + (−0.973 + 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (−0.927 − 0.373i)13-s + (−0.771 + 0.636i)14-s + (−0.264 + 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.582 + 0.813i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.582 + 0.813i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.582 + 0.813i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (41, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.582 + 0.813i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7818489422 + 0.4017832931i$
$L(\frac12,\chi)$  $\approx$  $0.7818489422 + 0.4017832931i$
$L(\chi,1)$  $\approx$  0.8787011823 + 0.2868654415i
$L(1,\chi)$  $\approx$  0.8787011823 + 0.2868654415i

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

−30.74362946936281721483173882523, −29.33002435315829995153076915714, −28.73182814665202264955763054497, −27.52501017108320144729983667354, −26.45980396117218141894882441513, −25.1513725773459223722265353470, −24.7838466717653571554638244873, −23.8374656100025895054734025160, −21.66459993562021341068846399763, −20.40461251308979243924962992735, −20.13469062365594607463695586275, −18.45547513702666005284597640689, −17.952233604378093920193583926203, −16.75072633469395595572875472509, −15.31970678128722990341910723652, −14.17238711691058914715604804629, −12.53730614369045383062360238322, −11.85745713368459892140029921158, −9.87569377130787113041138914971, −8.98313821473942670863347483057, −7.99331818488926163395743929074, −6.86282335846444399551658836600, −5.03307678424164713680053974733, −2.51837908716308409757869480444, −1.538611572250621689815957159144, 2.07015641029822920527783280948, 3.392947869863203481089240161285, 5.489997586257175042578277905914, 7.26371460393499074032987109195, 8.20069518989485546010463558252, 9.56227145328320212714014293898, 10.52367719552423438389889681238, 11.30339507074475179332148368334, 13.65850385062850356669536678233, 14.6741228013181309511991354684, 15.5893812825467078580360953146, 16.97834216927480337738288632344, 17.891935891210446607625136418426, 19.18089068882830550646497100202, 20.0404483341508172129417111228, 21.2472874853383234175607941891, 21.98290249669957494794423238497, 23.94685970296169689639278811152, 24.96887562608144518540202320525, 26.106908101485832185116710467927, 26.78743585126195346886724855987, 27.36178853916262945195672922489, 28.848650658494428107303519271217, 29.96122175344280527129571649935, 30.75835385429967551502913554124

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