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Properties

Label 1-83-83.38-r0-0-0
Degree $1$
Conductor $83$
Sign $0.930 - 0.365i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.930 - 0.365i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.930 - 0.365i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (-0.859 - 0.511i)T \)
3 \( 1 + (0.606 + 0.795i)T \)
5 \( 1 + (0.338 - 0.941i)T \)
7 \( 1 + (-0.409 - 0.912i)T \)
11 \( 1 + (0.953 + 0.301i)T \)
13 \( 1 + (0.896 + 0.443i)T \)
17 \( 1 + (0.190 - 0.981i)T \)
19 \( 1 + (-0.997 - 0.0765i)T \)
23 \( 1 + (0.720 + 0.693i)T \)
29 \( 1 + (0.988 + 0.152i)T \)
31 \( 1 + (0.0383 + 0.999i)T \)
37 \( 1 + (-0.264 - 0.964i)T \)
41 \( 1 + (-0.859 + 0.511i)T \)
43 \( 1 + (-0.927 - 0.373i)T \)
47 \( 1 + (-0.973 + 0.227i)T \)
53 \( 1 + (-0.973 - 0.227i)T \)
59 \( 1 + (0.817 - 0.575i)T \)
61 \( 1 + (-0.665 - 0.746i)T \)
67 \( 1 + (-0.543 + 0.839i)T \)
71 \( 1 + (-0.409 + 0.912i)T \)
73 \( 1 + (-0.771 + 0.636i)T \)
79 \( 1 + (0.477 + 0.878i)T \)
89 \( 1 + (-0.114 - 0.993i)T \)
97 \( 1 + (-0.114 + 0.993i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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