Properties

Degree 1
Conductor 139
Sign $0.997 - 0.0650i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.746 + 0.665i)2-s + (0.0227 − 0.999i)3-s + (0.113 + 0.993i)4-s + (0.983 − 0.181i)5-s + (0.682 − 0.730i)6-s + (0.291 − 0.956i)7-s + (−0.576 + 0.816i)8-s + (−0.998 − 0.0455i)9-s + (0.854 + 0.519i)10-s + (0.377 − 0.926i)11-s + (0.995 − 0.0909i)12-s + (−0.877 + 0.480i)13-s + (0.854 − 0.519i)14-s + (−0.158 − 0.987i)15-s + (−0.974 + 0.225i)16-s + (−0.829 + 0.557i)17-s + ⋯
L(s,χ)  = 1  + (0.746 + 0.665i)2-s + (0.0227 − 0.999i)3-s + (0.113 + 0.993i)4-s + (0.983 − 0.181i)5-s + (0.682 − 0.730i)6-s + (0.291 − 0.956i)7-s + (−0.576 + 0.816i)8-s + (−0.998 − 0.0455i)9-s + (0.854 + 0.519i)10-s + (0.377 − 0.926i)11-s + (0.995 − 0.0909i)12-s + (−0.877 + 0.480i)13-s + (0.854 − 0.519i)14-s + (−0.158 − 0.987i)15-s + (−0.974 + 0.225i)16-s + (−0.829 + 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.997 - 0.0650i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.997 - 0.0650i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $0.997 - 0.0650i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (67, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ 0.997 - 0.0650i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.764018122 - 0.05746979507i$
$L(\frac12,\chi)$  $\approx$  $1.764018122 - 0.05746979507i$
$L(\chi,1)$  $\approx$  1.631693849 + 0.03539159831i
$L(1,\chi)$  $\approx$  1.631693849 + 0.03539159831i

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

−28.589698502225437140434573373503, −27.705225430249558956277595273046, −26.65294579074243162104604146547, −25.17651179848906198207470193398, −24.69741856495843197722473766504, −22.96435364668094908556707801577, −22.03147568880109717231339926539, −21.789475919005097943288140440714, −20.55803635395504934820600776332, −19.93458206673036419302043050177, −18.328436323089442399957830970146, −17.45031160623636510325775785470, −15.87374332455747924214550948169, −14.86056220811992826998082923924, −14.32488012888497562678146423812, −12.92409129130793644115797027321, −11.825275079381929734177907681085, −10.73330244855448967216693686270, −9.67780254467778963571419327883, −9.0630470099719345402698016819, −6.700778852386188882461651514573, −5.276833528168657847193651965637, −4.825447883270329786949572985815, −3.02685546020306095013490995699, −2.12075272040848894053723211104, 1.62076926150603855346454728680, 3.16815388655212033954021693321, 4.83635701592190661835734566840, 6.04836376798159600778121965574, 6.90691291654650714367219172104, 7.97284024406675443021073342292, 9.21181370715086376552289738419, 11.042888402287862887512757668187, 12.18465292543091947642800451864, 13.42263482077659944855354178064, 13.79885486736005307891780873940, 14.71031181847466201404256658049, 16.52647927052569672218325736509, 17.19030656521620785402585227960, 17.928753148549991080543725173, 19.43603088323432299521397867471, 20.548370397717150128423839770887, 21.65900845445392291973615531759, 22.56181490118786199029825320207, 23.825586221831014847610136695967, 24.33250778407062752404803609836, 25.12329510268314525483173854055, 26.177151170852228785790182008701, 27.028802103088213053591950765194, 29.19763958924352603794781309064

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