Properties

Degree 1
Conductor 83
Sign $0.884 + 0.467i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0383 − 0.999i)2-s + (−0.927 + 0.373i)3-s + (−0.997 − 0.0765i)4-s + (−0.859 + 0.511i)5-s + (0.338 + 0.941i)6-s + (0.953 + 0.301i)7-s + (−0.114 + 0.993i)8-s + (0.720 − 0.693i)9-s + (0.477 + 0.878i)10-s + (0.606 + 0.795i)11-s + (0.953 − 0.301i)12-s + (0.190 + 0.981i)13-s + (0.338 − 0.941i)14-s + (0.606 − 0.795i)15-s + (0.988 + 0.152i)16-s + (−0.543 + 0.839i)17-s + ⋯
L(s,χ)  = 1  + (0.0383 − 0.999i)2-s + (−0.927 + 0.373i)3-s + (−0.997 − 0.0765i)4-s + (−0.859 + 0.511i)5-s + (0.338 + 0.941i)6-s + (0.953 + 0.301i)7-s + (−0.114 + 0.993i)8-s + (0.720 − 0.693i)9-s + (0.477 + 0.878i)10-s + (0.606 + 0.795i)11-s + (0.953 − 0.301i)12-s + (0.190 + 0.981i)13-s + (0.338 − 0.941i)14-s + (0.606 − 0.795i)15-s + (0.988 + 0.152i)16-s + (−0.543 + 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.884 + 0.467i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.884 + 0.467i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.884 + 0.467i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (9, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 83,\ (0:\ ),\ 0.884 + 0.467i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5607387887 + 0.1391113850i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5607387887 + 0.1391113850i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6835195149 - 0.04268497266i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6835195149 - 0.04268497266i\)

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

−30.81865579857375851945646276271, −29.93911666646473459742685216997, −28.312134247483050000030159869625, −27.37592763090395968599792402451, −26.962411472637220035385544948732, −24.93834418839093565787982182727, −24.3866593605853656493440696419, −23.439079561823480548151422902920, −22.70428554211996290994596180300, −21.41021156429664023166594071570, −19.74646743066805114272281319248, −18.42065951288060561275001077921, −17.492358415801338639228493709058, −16.53312672159568244026594594992, −15.66277376937018062231478682431, −14.27946325593767230079609042078, −12.98641932683982031994671268882, −11.84385487652836323255701717116, −10.63558490948247301566628908193, −8.59583212759537564428307010298, −7.76195822056278848471797984466, −6.46685240940636146461883117562, −5.14035848384346554170935191066, −4.15350091153315795958298079572, −0.77424717571091703907538488542, 1.80514165785037190761052381849, 3.9785462572337500030857921770, 4.67557994537996158823924455399, 6.503406997332214623428979926724, 8.29118671359250079238160952566, 9.76099513045479120031893664938, 11.1113207034933318003325502089, 11.557827445741989549493454931055, 12.60057692076471924510653061801, 14.4459975451353505292652907445, 15.35766203219481051686973384596, 17.07073578320364900289862312617, 17.97234753875584230413406682457, 19.0192090421793742330158386722, 20.19972330643288611922996814323, 21.515366556900151707387626272927, 22.1039121879387233547473093555, 23.38873946356657993701608690761, 23.893086631710202151763321722104, 26.11361038482120572507612589176, 27.26630980125624124709467255927, 27.82498155440978715936826312359, 28.64309900800717847517266019809, 30.04303493635595295119108613554, 30.69671943509491794383529936653

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