Properties

Degree 1
Conductor $ 2^{3} \cdot 7 $
Sign $0.386 + 0.922i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s − 15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s − 15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.386 + 0.922i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.386 + 0.922i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(56\)    =    \(2^{3} \cdot 7\)
\( \varepsilon \)  =  $0.386 + 0.922i$
motivic weight  =  \(0\)
character  :  $\chi_{56} (3, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 56,\ (0:\ ),\ 0.386 + 0.922i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7831311043 + 0.5209246280i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7831311043 + 0.5209246280i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9853529735 + 0.3949586307i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9853529735 + 0.3949586307i\)

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

−32.62226581537803238142745106992, −31.40354834054244039757311782718, −30.88534796706838815003140710171, −29.45440801738142322946192450052, −28.44834213985124964583053717317, −27.25379731544374331712185001285, −25.71538626442790863862629935448, −24.954033714979342472829084135097, −23.69484806656172175541247167264, −23.00688953255070787106684640005, −20.81798067798255846963686266251, −20.269692987560820180159614551004, −18.9201023277689602059161126016, −17.91090697953854904828455114495, −16.41450138176173723125088805570, −15.140616087304156406842556961449, −13.64193665242741608862021068466, −12.65598227578844444948413991532, −11.589505532440730239131325453577, −9.515195944154144649061963124194, −8.24030201410243590146529431667, −7.222322751380498019729854429015, −5.37105154696096913402046805253, −3.49915442338122875007197382990, −1.48445924299526608427941871755, 2.86592229947028444769445107760, 3.98274788399003067204061816101, 5.84367434213653341480433658441, 7.698173118292260187384135206786, 8.91619288649665427146502927957, 10.539992279648797561605690132891, 11.24474540088059881194093773060, 13.33677315490547045157807133177, 14.57460036805624808450531222602, 15.543349844025600373333509479349, 16.5811691152419186555591431359, 18.40955407094365035674089624407, 19.38365351217698016445250029531, 20.6940901781594864447242360442, 21.7569889886932054639833834159, 22.78749585396203842608140817548, 24.08505356587280379312737784967, 25.80149430434257479640600552011, 26.37753373264914409632099503886, 27.43366908793210430303059774234, 28.53313082223360704264976242599, 30.1945458082384610839988209397, 31.019623498325764102500489575255, 32.14484342853559572918264277117, 33.16506596328657197526284612113

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