Properties

Degree 1
Conductor 83
Sign $-0.0645 + 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.771 + 0.636i)2-s + (0.817 + 0.575i)3-s + (0.190 − 0.981i)4-s + (−0.973 − 0.227i)5-s + (−0.997 + 0.0765i)6-s + (0.720 + 0.693i)7-s + (0.477 + 0.878i)8-s + (0.338 + 0.941i)9-s + (0.896 − 0.443i)10-s + (−0.665 + 0.746i)11-s + (0.720 − 0.693i)12-s + (0.953 + 0.301i)13-s + (−0.997 − 0.0765i)14-s + (−0.665 − 0.746i)15-s + (−0.927 − 0.373i)16-s + (0.606 − 0.795i)17-s + ⋯
L(s,χ)  = 1  + (−0.771 + 0.636i)2-s + (0.817 + 0.575i)3-s + (0.190 − 0.981i)4-s + (−0.973 − 0.227i)5-s + (−0.997 + 0.0765i)6-s + (0.720 + 0.693i)7-s + (0.477 + 0.878i)8-s + (0.338 + 0.941i)9-s + (0.896 − 0.443i)10-s + (−0.665 + 0.746i)11-s + (0.720 − 0.693i)12-s + (0.953 + 0.301i)13-s + (−0.997 − 0.0765i)14-s + (−0.665 − 0.746i)15-s + (−0.927 − 0.373i)16-s + (0.606 − 0.795i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0645 + 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0645 + 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.0645 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (77, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ -0.0645 + 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5404492542 + 0.5765310567i$
$L(\frac12,\chi)$  $\approx$  $0.5404492542 + 0.5765310567i$
$L(\chi,1)$  $\approx$  0.7351142803 + 0.4339409430i
$L(1,\chi)$  $\approx$  0.7351142803 + 0.4339409430i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−30.34349543716782678497302966952, −29.94849949264037791352566305117, −28.3819117634213436704788862182, −27.305766718189313065792614232593, −26.43712592615040918454516405577, −25.71028564881729022314322260692, −24.14564484486100646178896302466, −23.4044481378032851689027533005, −21.53125590678585656402250896372, −20.49924731622918439394024557052, −19.74017222806127386031568267923, −18.735300321954127572474237050509, −17.962236970453601189037212562323, −16.44668768364700217602060719663, −15.175489380129390896535918615629, −13.73526515774019187062021404123, −12.6267263215043341262858603326, −11.31039855846221792447250791875, −10.39576994221521739382051682563, −8.39826087608884361996726682740, −8.15992239695028848062290236993, −6.85084500881427559118740021086, −4.03795885934323933340347800975, −2.973686664479458292313737661155, −1.17190879687712240283423513421, 2.07848542871376719742207135157, 4.14126108379850092447182184289, 5.47639655605151990412474826587, 7.57963645016326257254526228638, 8.20971129741316575178340300702, 9.27910725509577834957624188582, 10.57720593668821375707684914795, 11.87827531391496604290957473788, 13.83491199022743676231646596021, 15.097242420613877348156061245954, 15.58402235977571685695054787282, 16.64022264872084879005427504653, 18.32065058073489683837575009459, 19.016526820168596982346764192870, 20.35375027940794017371277818812, 20.979233564380558902751089819039, 22.87437701091842031371038281364, 23.922119477579706650725252972204, 25.04279197707645886509001229703, 25.86354367415171597887746299165, 26.95599152008897346255270399909, 27.79467584797900263414094622658, 28.34883429329039807947639783260, 30.34993162123360137644741856881, 31.51259582748972382207341894966

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