Properties

Label 1-83-83.45-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.674 + 0.738i$
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  + (0.859 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.674 + 0.738i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.674 + 0.738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190627826 + 2.701703913i\)
\(L(\frac12)\) \(\approx\) \(1.190627826 + 2.701703913i\)
\(L(1)\) \(\approx\) \(1.422416022 + 1.303980132i\)
\(L(1)\) \(\approx\) \(1.422416022 + 1.303980132i\)

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

−30.51594601873476134229251194483, −29.15111472844542517647512607292, −28.6118893909640142219867038056, −27.26349598483634921371422392946, −25.49381351695260610797132280096, −24.4800709256414030995755736193, −24.13798370883986925107063924592, −22.672208310863933486087019489866, −21.561973905244997024535445844267, −20.385881570901531773936924510481, −19.46034135929010792445497243088, −18.85467948607444152381093197844, −16.98956456413325461120846594776, −15.51061749759665674027097319618, −14.50067333954766343517305377531, −13.3228035070395106831249532899, −12.2501163180642180049638414412, −11.83415472830508323761921570241, −9.60530380827554693316163153644, −8.59449482649473331382325034444, −6.87025375928984258137322535572, −5.58647515334328137765028673573, −3.98871909635283908697027044948, −2.5507050098513907504744005248, −1.086249520778363380960601608811, 2.88637607302770311501198417840, 3.7294935965346957615149895356, 5.018160580152560026489749472913, 6.89771552828949544768061435618, 7.646258795327857433042390277104, 9.4876623286758008185243253067, 10.74754788991229773617634721761, 12.03773099841398710120824040013, 13.78692744594200478091995588376, 14.35220032059905132108601847584, 15.40298218164980078413829180246, 16.36271396364798942806310240498, 17.55331277468028003191039937488, 19.55950199579449735898795521509, 20.21656128450047312283319429860, 21.60967499889430383179827639473, 22.54441109065849329102358007232, 23.11124448633238648341257765705, 24.811607489912183894553152189803, 25.60240340558927067260969702690, 26.832704746768379788198915957693, 27.11315156957506612086512735673, 29.32251264491744650642689168389, 30.26451939226093076555833775243, 31.12850142545491899925853337208

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