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Properties

Label 1-83-83.44-r0-0-0
Degree $1$
Conductor $83$
Sign $-0.0622 - 0.998i$
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  + (−0.409 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0622 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0622 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.0622 - 0.998i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ -0.0622 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3291710390 - 0.3503337239i\)
\(L(\frac12)\) \(\approx\) \(0.3291710390 - 0.3503337239i\)
\(L(1)\) \(\approx\) \(0.6165713477 - 0.1309864623i\)
\(L(1)\) \(\approx\) \(0.6165713477 - 0.1309864623i\)

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

−31.36558600083155264046885980455, −30.16518378156832968623097164854, −28.785074081946505307074327722079, −28.00097237697659178690859714125, −26.85629236128701534463670646801, −26.33969752703448715756854136797, −25.234039125174950817280338246577, −23.092021893545311803678410781672, −22.420069495824724326966668876635, −21.35304752078775371256824525404, −20.09610999001900094938803140764, −19.5413261091668049531160019224, −18.471374747339067845267235191644, −16.87741745415544467445091124036, −15.74359963958257052192309353530, −14.73583789187693529175694183455, −13.06605399563336716595891661245, −12.02033421686590582408576593266, −10.542314529557848998946635710265, −9.92276369192862154417055064105, −8.56027371317455719180917047322, −7.43713599019426979675483993859, −4.93222189825168151357337696673, −3.54727482109897567383254305438, −2.706427602817470695624403791245, 0.56552978071490896469556786557, 3.10739313885499078314372160996, 5.00733919829827347462605726102, 6.72249373061897937190360932475, 7.49296993678078944714193319695, 8.61358676847443866440243633437, 9.72862992277259249330160108603, 11.66966568200991311983763617220, 13.068615170071015359910720091662, 13.9207378218190521546283080710, 15.38187414519511845095034037443, 16.217039318990482105626663966857, 17.42043855877108889316977397149, 18.94297456579837305215839295050, 19.21772962450005230472332443523, 20.43830657009887812821856524185, 22.55368896245864807915289233917, 23.48190238704795056941547416122, 24.28043983116876170230539664056, 25.15258137638110335833275471876, 26.444920269840439228576693506678, 26.90105985927701866430055833641, 28.61192020345484092798085786523, 29.26671200120783723426546147885, 30.99309532811072461628835560808

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