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Properties

Label 1-83-83.41-r0-0-0
Degree $1$
Conductor $83$
Sign $0.582 + 0.813i$
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  + (−0.997 + 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (0.477 + 0.878i)5-s + (−0.771 − 0.636i)6-s + (0.817 − 0.575i)7-s + (−0.973 + 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (−0.927 − 0.373i)13-s + (−0.771 + 0.636i)14-s + (−0.264 + 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (0.477 + 0.878i)5-s + (−0.771 − 0.636i)6-s + (0.817 − 0.575i)7-s + (−0.973 + 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (−0.927 − 0.373i)13-s + (−0.771 + 0.636i)14-s + (−0.264 + 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.582 + 0.813i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.582 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7818489422 + 0.4017832931i\)
\(L(\frac12)\) \(\approx\) \(0.7818489422 + 0.4017832931i\)
\(L(1)\) \(\approx\) \(0.8787011823 + 0.2868654415i\)
\(L(1)\) \(\approx\) \(0.8787011823 + 0.2868654415i\)

Euler product

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

−30.74362946936281721483173882523, −29.33002435315829995153076915714, −28.73182814665202264955763054497, −27.52501017108320144729983667354, −26.45980396117218141894882441513, −25.1513725773459223722265353470, −24.7838466717653571554638244873, −23.8374656100025895054734025160, −21.66459993562021341068846399763, −20.40461251308979243924962992735, −20.13469062365594607463695586275, −18.45547513702666005284597640689, −17.952233604378093920193583926203, −16.75072633469395595572875472509, −15.31970678128722990341910723652, −14.17238711691058914715604804629, −12.53730614369045383062360238322, −11.85745713368459892140029921158, −9.87569377130787113041138914971, −8.98313821473942670863347483057, −7.99331818488926163395743929074, −6.86282335846444399551658836600, −5.03307678424164713680053974733, −2.51837908716308409757869480444, −1.538611572250621689815957159144, 2.07015641029822920527783280948, 3.392947869863203481089240161285, 5.489997586257175042578277905914, 7.26371460393499074032987109195, 8.20069518989485546010463558252, 9.56227145328320212714014293898, 10.52367719552423438389889681238, 11.30339507074475179332148368334, 13.65850385062850356669536678233, 14.6741228013181309511991354684, 15.5893812825467078580360953146, 16.97834216927480337738288632344, 17.891935891210446607625136418426, 19.18089068882830550646497100202, 20.0404483341508172129417111228, 21.2472874853383234175607941891, 21.98290249669957494794423238497, 23.94685970296169689639278811152, 24.96887562608144518540202320525, 26.106908101485832185116710467927, 26.78743585126195346886724855987, 27.36178853916262945195672922489, 28.848650658494428107303519271217, 29.96122175344280527129571649935, 30.75835385429967551502913554124

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