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Properties

Label 1-83-83.9-r0-0-0
Degree $1$
Conductor $83$
Sign $0.884 + 0.467i$
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  + (0.0383 − 0.999i)2-s + (−0.927 + 0.373i)3-s + (−0.997 − 0.0765i)4-s + (−0.859 + 0.511i)5-s + (0.338 + 0.941i)6-s + (0.953 + 0.301i)7-s + (−0.114 + 0.993i)8-s + (0.720 − 0.693i)9-s + (0.477 + 0.878i)10-s + (0.606 + 0.795i)11-s + (0.953 − 0.301i)12-s + (0.190 + 0.981i)13-s + (0.338 − 0.941i)14-s + (0.606 − 0.795i)15-s + (0.988 + 0.152i)16-s + (−0.543 + 0.839i)17-s + ⋯
L(s)  = 1  + (0.0383 − 0.999i)2-s + (−0.927 + 0.373i)3-s + (−0.997 − 0.0765i)4-s + (−0.859 + 0.511i)5-s + (0.338 + 0.941i)6-s + (0.953 + 0.301i)7-s + (−0.114 + 0.993i)8-s + (0.720 − 0.693i)9-s + (0.477 + 0.878i)10-s + (0.606 + 0.795i)11-s + (0.953 − 0.301i)12-s + (0.190 + 0.981i)13-s + (0.338 − 0.941i)14-s + (0.606 − 0.795i)15-s + (0.988 + 0.152i)16-s + (−0.543 + 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.884 + 0.467i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.884 + 0.467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5607387887 + 0.1391113850i\)
\(L(\frac12)\) \(\approx\) \(0.5607387887 + 0.1391113850i\)
\(L(1)\) \(\approx\) \(0.6835195149 - 0.04268497266i\)
\(L(1)\) \(\approx\) \(0.6835195149 - 0.04268497266i\)

Euler product

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

−30.81865579857375851945646276271, −29.93911666646473459742685216997, −28.312134247483050000030159869625, −27.37592763090395968599792402451, −26.962411472637220035385544948732, −24.93834418839093565787982182727, −24.3866593605853656493440696419, −23.439079561823480548151422902920, −22.70428554211996290994596180300, −21.41021156429664023166594071570, −19.74646743066805114272281319248, −18.42065951288060561275001077921, −17.492358415801338639228493709058, −16.53312672159568244026594594992, −15.66277376937018062231478682431, −14.27946325593767230079609042078, −12.98641932683982031994671268882, −11.84385487652836323255701717116, −10.63558490948247301566628908193, −8.59583212759537564428307010298, −7.76195822056278848471797984466, −6.46685240940636146461883117562, −5.14035848384346554170935191066, −4.15350091153315795958298079572, −0.77424717571091703907538488542, 1.80514165785037190761052381849, 3.9785462572337500030857921770, 4.67557994537996158823924455399, 6.503406997332214623428979926724, 8.29118671359250079238160952566, 9.76099513045479120031893664938, 11.1113207034933318003325502089, 11.557827445741989549493454931055, 12.60057692076471924510653061801, 14.4459975451353505292652907445, 15.35766203219481051686973384596, 17.07073578320364900289862312617, 17.97234753875584230413406682457, 19.0192090421793742330158386722, 20.19972330643288611922996814323, 21.515366556900151707387626272927, 22.1039121879387233547473093555, 23.38873946356657993701608690761, 23.893086631710202151763321722104, 26.11361038482120572507612589176, 27.26630980125624124709467255927, 27.82498155440978715936826312359, 28.64309900800717847517266019809, 30.04303493635595295119108613554, 30.69671943509491794383529936653

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