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  :  $\chi_{139} (30, \cdot )$
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

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

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