Properties

Degree $1$
Conductor $83$
Sign $0.930 - 0.365i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.859 − 0.511i)2-s + (0.606 + 0.795i)3-s + (0.477 + 0.878i)4-s + (0.338 − 0.941i)5-s + (−0.114 − 0.993i)6-s + (−0.409 − 0.912i)7-s + (0.0383 − 0.999i)8-s + (−0.264 + 0.964i)9-s + (−0.771 + 0.636i)10-s + (0.953 + 0.301i)11-s + (−0.409 + 0.912i)12-s + (0.896 + 0.443i)13-s + (−0.114 + 0.993i)14-s + (0.953 − 0.301i)15-s + (−0.543 + 0.839i)16-s + (0.190 − 0.981i)17-s + ⋯
L(s,χ)  = 1  + (−0.859 − 0.511i)2-s + (0.606 + 0.795i)3-s + (0.477 + 0.878i)4-s + (0.338 − 0.941i)5-s + (−0.114 − 0.993i)6-s + (−0.409 − 0.912i)7-s + (0.0383 − 0.999i)8-s + (−0.264 + 0.964i)9-s + (−0.771 + 0.636i)10-s + (0.953 + 0.301i)11-s + (−0.409 + 0.912i)12-s + (0.896 + 0.443i)13-s + (−0.114 + 0.993i)14-s + (0.953 − 0.301i)15-s + (−0.543 + 0.839i)16-s + (0.190 − 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.930 - 0.365i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.930 - 0.365i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.930 - 0.365i$
Motivic weight: \(0\)
Character: $\chi_{83} (38, \cdot )$
Sato-Tate group: $\mu(41)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.930 - 0.365i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.8284545687 - 0.1569137263i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.8284545687 - 0.1569137263i\)
\(L(\chi,1)\) \(\approx\) \(0.8863160301 - 0.1149247595i\)
\(L(1,\chi)\) \(\approx\) \(0.8863160301 - 0.1149247595i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.6865347211182201686707846134, −29.86165249743207199514135562556, −28.83195262203088038829462752044, −27.65082255051765202058657572426, −26.387612363215750239995078883524, −25.468501385342954030768154004107, −25.083356001144208028852888406537, −23.75577725612191294990788721912, −22.56915471064186519231028450406, −21.04915614126940022387281227809, −19.49055456507099899031565524972, −18.91775022771256877019585935613, −18.09297538468444440134978633931, −16.97148145536815409674820875270, −15.24211343807081541941978264686, −14.687355221613104177358660331320, −13.342536118151844158799717247159, −11.76418031006156841782417894350, −10.372882574580528421612452403562, −9.00072336369753331486026623533, −8.17945708781522495487030192084, −6.49583809412489378257352402417, −6.197487664601867955645505483720, −3.114426284616928924983249849415, −1.73785948284241891706597648708, 1.49899238978799929824132095013, 3.403415146834937748357378363728, 4.54325829449381382048550706836, 6.82569174809001197594338838614, 8.4349799239464733028410663617, 9.26003666814773112881735007015, 10.15020024024779880026162461878, 11.42653202155048730241225665709, 12.96381845864159745724188151057, 14.0366939575802778733055197613, 15.87515109799105082968924695569, 16.60732200213054531055655021980, 17.49711527555476516428141897130, 19.27001707868982476584979251899, 20.03390218613513687500351139232, 20.84961007024976229739135859162, 21.68005984075781301028933265697, 23.217646508601697156293424286718, 25.09009617394740562460691133159, 25.52836254988520366357397813214, 26.79898478312650712562425400548, 27.57392459915070623191846100021, 28.49217581658478108109895208157, 29.59577547138028421150757757482, 30.68981172557789931521211400639

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