Properties

Degree 1
Conductor 83
Sign $0.961 + 0.275i$
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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.961 + 0.275i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.961 + 0.275i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.961 + 0.275i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (14, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.961 + 0.275i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.705323914 + 0.5201633161i$
$L(\frac12,\chi)$  $\approx$  $3.705323914 + 0.5201633161i$
$L(\chi,1)$  $\approx$  2.314609725 + 0.3136703441i
$L(1,\chi)$  $\approx$  2.314609725 + 0.3136703441i

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

−30.65526552503910410904447654473, −29.78201906910213477307678563729, −28.48530256500043556203371574629, −27.63598612451540145791895088539, −26.24078812116103655622425568617, −25.06171477814813753416658640744, −24.35658550247963134666407053211, −22.60420417855768544163149040236, −21.74407763656696210797581045225, −20.97100916547195613228745469281, −20.17307771838898562582314156844, −18.80135755920828755042658648945, −17.70553476694481505388193428271, −15.71563622821724456511537385771, −14.80401333358102513021239629900, −13.9954135052514776799015933613, −12.90352135003952958561936406000, −11.464186302014350736641203856839, −10.01410813943556686175991145834, −9.464238280358735950690130940451, −7.56769581602420905797891658060, −5.53016797529081330489524859772, −4.70519559328460378385229996114, −2.802583063564380145021604065096, −2.05698022381148407598308827990, 1.76235476668178509465326616204, 3.28540613912011374444019740374, 4.935327437326465158786175855535, 6.28710632326313231577999360314, 7.62994402796137601144195648443, 8.507837295939345038707350223169, 10.18442111791458000076314682241, 12.102644264316956019416765677590, 13.19123236846154889798328491757, 14.11385655947735305942320392523, 14.6422527431121786980377562659, 16.38836867135334522115889812862, 17.40514170001999365459703961715, 18.49358608206732867506627574592, 20.2103105958882577504196307617, 21.03294827697213086837589650788, 21.91835406659753918284152696943, 23.66521568493497561675290717528, 24.222297169260868507935406441277, 25.13681134686273227636691040958, 26.167853659328190959927626925991, 26.92229273800294908942056040497, 29.10769488393176173283798854327, 29.83549641281594810801794848673, 30.73852198684283500856458474063

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