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Properties

Label 1-83-83.5-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.840 - 0.542i$
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  + (−0.477 + 0.878i)2-s + (−0.264 − 0.964i)3-s + (−0.543 − 0.839i)4-s + (0.771 − 0.636i)5-s + (0.973 + 0.227i)6-s + (−0.665 − 0.746i)7-s + (0.997 − 0.0765i)8-s + (−0.859 + 0.511i)9-s + (0.190 + 0.981i)10-s + (0.817 − 0.575i)11-s + (−0.665 + 0.746i)12-s + (−0.606 + 0.795i)13-s + (0.973 − 0.227i)14-s + (−0.817 − 0.575i)15-s + (−0.409 + 0.912i)16-s + (−0.927 + 0.373i)17-s + ⋯
L(s)  = 1  + (−0.477 + 0.878i)2-s + (−0.264 − 0.964i)3-s + (−0.543 − 0.839i)4-s + (0.771 − 0.636i)5-s + (0.973 + 0.227i)6-s + (−0.665 − 0.746i)7-s + (0.997 − 0.0765i)8-s + (−0.859 + 0.511i)9-s + (0.190 + 0.981i)10-s + (0.817 − 0.575i)11-s + (−0.665 + 0.746i)12-s + (−0.606 + 0.795i)13-s + (0.973 − 0.227i)14-s + (−0.817 − 0.575i)15-s + (−0.409 + 0.912i)16-s + (−0.927 + 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.840 - 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1578138142 - 0.5355270209i\)
\(L(\frac12)\) \(\approx\) \(0.1578138142 - 0.5355270209i\)
\(L(1)\) \(\approx\) \(0.6240747891 - 0.1693289321i\)
\(L(1)\) \(\approx\) \(0.6240747891 - 0.1693289321i\)

Euler product

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

−30.95886780002887026584766468864, −29.61161851912314249722064915097, −28.97237155018246077485718437257, −27.83763043276057856100314084967, −27.10878117893540894043245989267, −25.83468424152616588931794588955, −25.288999530662053533619472764877, −22.82330400097772651634069967776, −22.1308138049885438637321721868, −21.569609688358520623954076607716, −20.25154271509993094769880411415, −19.25514188594825257837702767213, −17.785659445910432776548427178218, −17.23736079179610280229798541731, −15.69822068837188322763937830291, −14.50840552199998896972430632667, −12.96030061515119695206988232145, −11.73825685659546285538639090493, −10.53791764427176780970580380001, −9.669107727842619876009178313982, −8.87300841345239453680426584155, −6.71023958046720511213955444918, −5.12205335202899631467497598481, −3.49196417235742540678072091702, −2.29090703504199231744770241388, 0.30946779166315423932375272526, 1.77176701336330068817956063502, 4.58171141734930812407452394290, 6.266113702469928594948887661678, 6.70866517148114511516563318477, 8.34265128499479142507599545329, 9.36226316821814722196856419716, 10.78727462263045940658793164592, 12.611575553517269178975474398047, 13.59816383157665702594474723391, 14.44295043847983407727638925402, 16.5010784454802304289697344233, 16.8996312175034119899318067736, 17.886356370651535641333706583933, 19.231069476992137989554146135837, 19.881397911162342926040234657421, 21.86324094574612924659048812265, 23.03705040214890759089286884834, 24.12381025362066845598413994786, 24.72709360851899738472609053289, 25.74588210652961064095582866192, 26.752736455862280443891546258409, 28.22510064513022730376694811519, 29.055400786793409522889664622027, 29.82904836149964603153154964003

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