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Properties

Label 1-83-83.29-r0-0-0
Degree $1$
Conductor $83$
Sign $0.0614 + 0.998i$
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  + (0.606 + 0.795i)2-s + (−0.973 − 0.227i)3-s + (−0.264 + 0.964i)4-s + (0.953 − 0.301i)5-s + (−0.409 − 0.912i)6-s + (0.477 + 0.878i)7-s + (−0.927 + 0.373i)8-s + (0.896 + 0.443i)9-s + (0.817 + 0.575i)10-s + (−0.997 − 0.0765i)11-s + (0.477 − 0.878i)12-s + (−0.114 + 0.993i)13-s + (−0.409 + 0.912i)14-s + (−0.997 + 0.0765i)15-s + (−0.859 − 0.511i)16-s + (0.338 + 0.941i)17-s + ⋯
L(s)  = 1  + (0.606 + 0.795i)2-s + (−0.973 − 0.227i)3-s + (−0.264 + 0.964i)4-s + (0.953 − 0.301i)5-s + (−0.409 − 0.912i)6-s + (0.477 + 0.878i)7-s + (−0.927 + 0.373i)8-s + (0.896 + 0.443i)9-s + (0.817 + 0.575i)10-s + (−0.997 − 0.0765i)11-s + (0.477 − 0.878i)12-s + (−0.114 + 0.993i)13-s + (−0.409 + 0.912i)14-s + (−0.997 + 0.0765i)15-s + (−0.859 − 0.511i)16-s + (0.338 + 0.941i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0614 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0614 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.0614 + 0.998i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.0614 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7895317864 + 0.7424470195i\)
\(L(\frac12)\) \(\approx\) \(0.7895317864 + 0.7424470195i\)
\(L(1)\) \(\approx\) \(0.9975911307 + 0.5587075603i\)
\(L(1)\) \(\approx\) \(0.9975911307 + 0.5587075603i\)

Euler product

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

−30.251797905340018083881929299904, −29.462291903563018525996423041, −28.95155209956143248204349069667, −27.61720691064525064015142028684, −26.80969194611160712499113519319, −25.074047212236709390142246190223, −23.72658419651706267111894692741, −22.99227146183413127886806869791, −22.00464033271798999248724724750, −21.03571397662021660598692607091, −20.25484137897232134236280010858, −18.292091484876042174503375088988, −17.89626303120461330094831512461, −16.40108277795064491158493016353, −14.93338848867777083170902646402, −13.6834795394652609576263513851, −12.780895902144381518744266294629, −11.35505675975029442551953972872, −10.42466777647082708474971885151, −9.76542353764776577502103838723, −7.26811552572106525269699567040, −5.63515958486598203822476256850, −4.99115801727501699782245045092, −3.210851053244489387292216241540, −1.311350878960791353025002364376, 2.234323867643562009160844540027, 4.678330477860464171132922246865, 5.552763873224556349896920833574, 6.45345789679621682681852938987, 7.97951916178385709459947103137, 9.43372768223609974378785607145, 11.17096703130684239573885474189, 12.43151932988415095673816570485, 13.23820958151084314419460362281, 14.554419590122497685574797987190, 15.890153782450281360098695741233, 16.83687880334145824636311627050, 17.831298174808337759780401068279, 18.59689530139430400303656169381, 21.08839164223398411615772710109, 21.567563140947550479887127869938, 22.565322406625766258922143462066, 23.98476692854364021882195667726, 24.34507404811009144085575946877, 25.57333917358622361013257952252, 26.6698865092407965315231191865, 28.26773268740280062298081096118, 28.880170642231846366995121421165, 30.218251237693191163283225237721, 31.17328133797584968157802220727

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