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Properties

Label 1-83-83.52-r1-0-0
Degree $1$
Conductor $83$
Sign $0.999 - 0.0357i$
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  + (0.973 − 0.227i)2-s + (−0.665 + 0.746i)3-s + (0.896 − 0.443i)4-s + (0.997 + 0.0765i)5-s + (−0.477 + 0.878i)6-s + (−0.264 − 0.964i)7-s + (0.771 − 0.636i)8-s + (−0.114 − 0.993i)9-s + (0.988 − 0.152i)10-s + (0.720 + 0.693i)11-s + (−0.264 + 0.964i)12-s + (0.409 + 0.912i)13-s + (−0.477 − 0.878i)14-s + (−0.720 + 0.693i)15-s + (0.606 − 0.795i)16-s + (0.953 − 0.301i)17-s + ⋯
L(s)  = 1  + (0.973 − 0.227i)2-s + (−0.665 + 0.746i)3-s + (0.896 − 0.443i)4-s + (0.997 + 0.0765i)5-s + (−0.477 + 0.878i)6-s + (−0.264 − 0.964i)7-s + (0.771 − 0.636i)8-s + (−0.114 − 0.993i)9-s + (0.988 − 0.152i)10-s + (0.720 + 0.693i)11-s + (−0.264 + 0.964i)12-s + (0.409 + 0.912i)13-s + (−0.477 − 0.878i)14-s + (−0.720 + 0.693i)15-s + (0.606 − 0.795i)16-s + (0.953 − 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.999 - 0.0357i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.999 - 0.0357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.973208408 - 0.05309153821i\)
\(L(\frac12)\) \(\approx\) \(2.973208408 - 0.05309153821i\)
\(L(1)\) \(\approx\) \(1.908268340 + 0.01349561648i\)
\(L(1)\) \(\approx\) \(1.908268340 + 0.01349561648i\)

Euler product

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

−30.33487224132279718413082917771, −29.81516877330753145695920991240, −28.82769493782076239641479577785, −27.84814664597657552919365434605, −25.64903705103990152126435623745, −25.126624535558243386321706483288, −24.24434695733207318916982230603, −23.0901572573830878432030534858, −21.9731629424758704929982566979, −21.52405724951079803273533472109, −19.847078796051645348384606669099, −18.50900679693267573965209574949, −17.33105210256142202632311840775, −16.405075682562330524283581062819, −14.994773451911610164045181388961, −13.66229207384481521316977719865, −12.89564206238601557066020022464, −11.87484291573287236374551295314, −10.680368009041089493370505361113, −8.755224777804115039000023941654, −7.08039922146219402407355655753, −5.81144207619431534441978763546, −5.44735333656085038171739444127, −3.10092291719965498103891906667, −1.594725537207360282347421405118, 1.49424387443532827278191121875, 3.57669649390367554429146514695, 4.65260072262297441153546965361, 6.003705834185768994640098208146, 6.89951893266405455327316303528, 9.597106789487603901914409226512, 10.32227964503330027277949753125, 11.52054948909007493465946988158, 12.750153550514575501975281755143, 14.05631644986247265729515977372, 14.83551799871569545390473936907, 16.57357698991859585345211391553, 16.89350384910828506669748597022, 18.6771470332179210489874124637, 20.468204607645597018604330715310, 20.92193792240810548529649227808, 22.164717300921636147903810316796, 22.84126905094692185547491075141, 23.81795636422831726447413311324, 25.25201725401018937804548373437, 26.22127726450686666664176689170, 27.696159198102890331501155738788, 28.81823519651232740691220940318, 29.51050891533872888724332743445, 30.44137513001086227897968314359

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