Properties

Degree 1
Conductor 139
Sign $0.661 + 0.750i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.898 + 0.439i)2-s + (0.983 − 0.181i)3-s + (0.613 + 0.789i)4-s + (0.113 + 0.993i)5-s + (0.962 + 0.269i)6-s + (−0.715 − 0.699i)7-s + (0.203 + 0.979i)8-s + (0.934 − 0.356i)9-s + (−0.334 + 0.942i)10-s + (−0.998 − 0.0455i)11-s + (0.746 + 0.665i)12-s + (−0.648 − 0.761i)13-s + (−0.334 − 0.942i)14-s + (0.291 + 0.956i)15-s + (−0.247 + 0.968i)16-s + (0.0227 − 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.898 + 0.439i)2-s + (0.983 − 0.181i)3-s + (0.613 + 0.789i)4-s + (0.113 + 0.993i)5-s + (0.962 + 0.269i)6-s + (−0.715 − 0.699i)7-s + (0.203 + 0.979i)8-s + (0.934 − 0.356i)9-s + (−0.334 + 0.942i)10-s + (−0.998 − 0.0455i)11-s + (0.746 + 0.665i)12-s + (−0.648 − 0.761i)13-s + (−0.334 − 0.942i)14-s + (0.291 + 0.956i)15-s + (−0.247 + 0.968i)16-s + (0.0227 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.661 + 0.750i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.661 + 0.750i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $0.661 + 0.750i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (51, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ 0.661 + 0.750i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.993701086 + 0.9000447968i$
$L(\frac12,\chi)$  $\approx$  $1.993701086 + 0.9000447968i$
$L(\chi,1)$  $\approx$  1.895548210 + 0.5961960944i
$L(1,\chi)$  $\approx$  1.895548210 + 0.5961960944i

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

−28.605001343650373737208885496366, −27.51530684108949903446137501403, −25.97713062919968283131994914687, −25.2546147881959199697552257124, −24.22933333409941547286013848621, −23.52213346765864267044640353637, −21.84765814983525228994356932257, −21.38740315820656589521558991249, −20.44172675685614818466785339313, −19.440168677302216380814289484668, −18.84161041450468234776490237462, −16.812329775344767731807433440180, −15.62534090724250459554274823910, −15.07143468769064376207510927748, −13.65174798437939299583555180775, −12.92223064628146131140118348138, −12.193204522017538321998960615141, −10.46586105258420301439080036314, −9.44386731919021078569810959980, −8.44139938464758074967783000227, −6.80238301790951871011691927201, −5.30139809679657332228390589089, −4.31958732713607719039084112011, −2.92630993041758036513704016188, −1.88376697205674392253354516039, 2.61461183233135575195986290192, 3.13056377536281939006265302866, 4.56254303929839450131792823044, 6.27359449770045391368018602871, 7.26579538781708956244093482095, 7.99622247066305552295468473943, 9.81620402739185762458989406639, 10.81794237735755808628298188103, 12.53382654149642199526221436350, 13.36311852705991750655461724650, 14.17935780440580850779283668586, 15.131430754020531992761789055379, 15.90806976491255614758339867250, 17.34095077629724954023130500920, 18.63284109376743328117885694068, 19.68113417706597686189039814189, 20.710148361327456068050444354371, 21.5909644584735972795080165862, 22.87786138113885028335125873318, 23.33908478279692407336480369912, 24.86620870909473098146555077843, 25.395483560675100455230043193602, 26.551907185767240740860342532350, 26.79011640738624079182830485667, 29.17843798760508025170562000246

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