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Properties

Label 1-83-83.66-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.998 + 0.0559i$
Analytic cond. $8.91958$
Root an. cond. $8.91958$
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  + (0.409 + 0.912i)2-s + (0.477 + 0.878i)3-s + (−0.665 + 0.746i)4-s + (0.927 − 0.373i)5-s + (−0.606 + 0.795i)6-s + (−0.973 + 0.227i)7-s + (−0.953 − 0.301i)8-s + (−0.543 + 0.839i)9-s + (0.720 + 0.693i)10-s + (−0.771 + 0.636i)11-s + (−0.973 − 0.227i)12-s + (0.859 + 0.511i)13-s + (−0.606 − 0.795i)14-s + (0.771 + 0.636i)15-s + (−0.114 − 0.993i)16-s + (0.0383 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.409 + 0.912i)2-s + (0.477 + 0.878i)3-s + (−0.665 + 0.746i)4-s + (0.927 − 0.373i)5-s + (−0.606 + 0.795i)6-s + (−0.973 + 0.227i)7-s + (−0.953 − 0.301i)8-s + (−0.543 + 0.839i)9-s + (0.720 + 0.693i)10-s + (−0.771 + 0.636i)11-s + (−0.973 − 0.227i)12-s + (0.859 + 0.511i)13-s + (−0.606 − 0.795i)14-s + (0.771 + 0.636i)15-s + (−0.114 − 0.993i)16-s + (0.0383 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.998 + 0.0559i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.998 + 0.0559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05534961816 + 1.978068603i\)
\(L(\frac12)\) \(\approx\) \(0.05534961816 + 1.978068603i\)
\(L(1)\) \(\approx\) \(0.8335544702 + 1.124392697i\)
\(L(1)\) \(\approx\) \(0.8335544702 + 1.124392697i\)

Euler product

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

−29.83050413143680804447719451944, −29.38184775162455223281911865596, −28.57343477015451465632300013077, −26.83309665259768272991527588386, −25.75337658626745624291849812500, −24.803238347991754995412654085921, −23.32794084466718228538582783228, −22.69918550458880134832127635952, −21.21001606000930999022471596267, −20.48092402427894381396174734025, −19.02542334357243733245265420745, −18.618087573147046980021970541170, −17.358086052584285287713802890561, −15.41521120931030010123993238155, −13.82867108794741535950745834423, −13.47464229853905200701385744848, −12.4664329474299702003552983607, −10.88812489470269764264956668427, −9.79431388455814037988312572804, −8.551881123571213247418719076486, −6.63409275233973510434089763882, −5.62693533962236274410605564602, −3.34774284299989966411637770482, −2.480061201795149186082850792951, −0.78164668315724642541374769581, 2.7126206967791472244662059017, 4.22999328967832849822596374545, 5.461433021809974990828846124157, 6.62988412532143723787404042456, 8.4941685074694431556647519254, 9.28581688729608848642223277899, 10.470679870511257727603867447207, 12.76190833459240468621010703962, 13.43979564936025012013497816933, 14.72395039660374664024085885028, 15.73294991384788805048273892675, 16.58060162611383119389745504585, 17.65464739401074570577231809090, 19.17134639903005649327591992249, 20.85047968253576771935047853979, 21.46456839964917737667443590803, 22.51095080146738706180984181456, 23.62908203625135296001759467113, 25.12093325343534309613331120764, 25.84018675971985109077188531429, 26.2789510936671146896434294913, 27.92972065171304962254312129046, 28.76984618968998950740326312915, 30.46323322113114835158579021762, 31.540132409427233164383770033791

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