Properties

Degree 1
Conductor 61
Sign $0.183 - 0.982i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.207 + 0.978i)2-s + (−0.309 + 0.951i)3-s + (−0.913 + 0.406i)4-s + (0.104 − 0.994i)5-s + (−0.994 − 0.104i)6-s + (−0.743 − 0.669i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.994 − 0.104i)10-s + i·11-s + (−0.104 − 0.994i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.913 + 0.406i)15-s + (0.669 − 0.743i)16-s + (0.406 + 0.913i)17-s + ⋯
L(s,χ)  = 1  + (0.207 + 0.978i)2-s + (−0.309 + 0.951i)3-s + (−0.913 + 0.406i)4-s + (0.104 − 0.994i)5-s + (−0.994 − 0.104i)6-s + (−0.743 − 0.669i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.994 − 0.104i)10-s + i·11-s + (−0.104 − 0.994i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.913 + 0.406i)15-s + (0.669 − 0.743i)16-s + (0.406 + 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.183 - 0.982i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.183 - 0.982i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.183 - 0.982i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (18, \cdot )$
Sato-Tate  :  $\mu(60)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 61,\ (1:\ ),\ 0.183 - 0.982i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1336185273 - 0.1109673565i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1336185273 - 0.1109673565i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5701189762 + 0.3166997891i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5701189762 + 0.3166997891i\)

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

−31.96414445747699878740643991268, −31.26668540234283272898094106867, −29.988032622282161394796727099028, −29.407965346188397342681745248613, −28.55815212893030206198620051738, −27.11471260931550599586725103970, −25.88922499004758635162486173653, −24.48284466109770582040578824796, −23.17781144889241122956950400227, −22.35267927481512998219862666481, −21.44002977497455218493114240620, −19.61131149458073349552832749922, −18.776068539536242607097527608896, −18.301442537749800825288023093326, −16.6002598520464949586929966357, −14.537268010710074631177375643738, −13.66561489312877153099070439154, −12.33045885568320788010402310950, −11.47760113910957694555745179406, −10.18485018624091653433701574904, −8.63930040107530066391018488142, −6.75313290058799439705315937561, −5.59353861156279606267522083743, −3.27275789783716310717692570797, −2.10507786765962107569849757401, 0.0864100535052616952583504307, 3.79607469271373463366713328838, 4.85772001901585284866501234682, 6.07870741998025174211209683035, 7.78495160162771906424914971694, 9.29451141823276885172174721340, 10.14133861103178060774186731335, 12.33155544411206586622769633265, 13.32862396496721478882767489349, 14.982616658111124075646471501324, 15.79892754466055410219646623233, 17.05754136636647042636252120821, 17.39013560706993942128828744848, 19.68786929967967567317623019130, 20.84686778227909130008971624244, 22.126907579538458001421168472840, 23.05875448558269953940399863102, 24.04344451055218423759592886073, 25.54286247414737399607875742386, 26.188214679592200637030534403982, 27.64701834129467563750376046876, 28.18622520843217082398543034047, 29.81063028716239459840217261023, 31.52556514363732635195812374689, 32.44253934146727130344936197468

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