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Properties

Label 1-83-83.61-r0-0-0
Degree $1$
Conductor $83$
Sign $-0.400 - 0.916i$
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  + (0.338 − 0.941i)2-s + (0.953 − 0.301i)3-s + (−0.771 − 0.636i)4-s + (−0.114 − 0.993i)5-s + (0.0383 − 0.999i)6-s + (−0.927 + 0.373i)7-s + (−0.859 + 0.511i)8-s + (0.817 − 0.575i)9-s + (−0.973 − 0.227i)10-s + (−0.409 + 0.912i)11-s + (−0.927 − 0.373i)12-s + (0.988 − 0.152i)13-s + (0.0383 + 0.999i)14-s + (−0.409 − 0.912i)15-s + (0.190 + 0.981i)16-s + (0.896 + 0.443i)17-s + ⋯
L(s)  = 1  + (0.338 − 0.941i)2-s + (0.953 − 0.301i)3-s + (−0.771 − 0.636i)4-s + (−0.114 − 0.993i)5-s + (0.0383 − 0.999i)6-s + (−0.927 + 0.373i)7-s + (−0.859 + 0.511i)8-s + (0.817 − 0.575i)9-s + (−0.973 − 0.227i)10-s + (−0.409 + 0.912i)11-s + (−0.927 − 0.373i)12-s + (0.988 − 0.152i)13-s + (0.0383 + 0.999i)14-s + (−0.409 − 0.912i)15-s + (0.190 + 0.981i)16-s + (0.896 + 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7034769703 - 1.075590447i\)
\(L(\frac12)\) \(\approx\) \(0.7034769703 - 1.075590447i\)
\(L(1)\) \(\approx\) \(1.014555961 - 0.8603500275i\)
\(L(1)\) \(\approx\) \(1.014555961 - 0.8603500275i\)

Euler product

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

−31.39582847284904919696867357897, −30.50767776643041730962784938737, −29.44239538243925046808138294884, −27.44204026648957553696098569036, −26.55311250710046243489077133477, −25.924757339173854641816563409065, −25.138179921127196004014231503431, −23.71581682979417254742564804826, −22.710489630182569848215228311149, −21.75725659896271319203039414382, −20.5757128436244245190272056206, −18.9237621777247487745821931504, −18.43078594261783672160326889028, −16.338595292806748963022576570691, −15.92799523133937275249836534150, −14.45276398300127247193823757214, −13.928892472913395485323596339958, −12.746638197850272721741708539530, −10.67860870545693084874580850305, −9.438592954496326402004603793473, −8.124555839043009445614021778024, −7.0786425162828045649772521734, −5.8164580055529499355125918559, −3.757214690609935909394061632129, −3.13751188428034232436055507486, 1.46558707201242138283941363422, 2.99689835097394808026368807928, 4.21050363061981514452605450240, 5.82315863292592141317381032528, 7.87484622621011855819255299532, 9.20011576642551976554983500749, 9.8331092304528788552774840148, 11.76647812086458224722604537606, 13.032122143557924368349829009251, 13.21401857198216082596260109451, 14.91324535610540768223266252756, 15.95126192599978559472783557645, 17.84738363518847331345180168157, 18.97278058994737098889064180711, 19.896952875850469168735073881141, 20.62265082309940867054106436631, 21.58141729670586588491890686767, 23.08694028069024286882219177941, 23.96181107252800011081111838528, 25.30362435292871341592222084312, 26.1675770350849343575311891346, 27.806155488608860145257164824533, 28.43528793224966832816427216526, 29.586911206056327897207828099094, 30.69635597321293901370982147811

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