Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $0.0633 - 0.997i$
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)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s − 13-s + 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + 20-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s − 13-s + 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + 20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0633 - 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0633 - 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $0.0633 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (65, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 77,\ (1:\ ),\ 0.0633 - 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.02398348336 + 0.02250973164i$
$L(\frac12,\chi)$  $\approx$  $0.02398348336 + 0.02250973164i$
$L(\chi,1)$  $\approx$  0.5961857215 + 0.3965719450i
$L(1,\chi)$  $\approx$  0.5961857215 + 0.3965719450i

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

−30.94883885679734357989920827522, −30.112557231579447643975463534239, −29.45815231286545183368219289741, −28.280605390604500151153350750481, −27.32733088588348226194496746251, −25.92853661279018772841389024673, −24.23346215169183134371519098836, −23.6068427904832031595421651446, −22.427799554378269105097713501208, −21.862309342021910643441954579802, −20.03578933837683324455379388635, −19.21382844604431397394167601267, −18.381875737866582344845470960874, −17.21721972698921281534505020460, −15.2829557162154334339151841511, −14.196177946329940429022865510179, −13.019018842228865372990556712191, −11.88323204496605895086883380662, −11.13568559224866176033538661932, −9.85719430819394854935732867914, −7.833672125283971141665338417282, −6.52586381500112728642611169265, −5.15032142446298601093268385897, −3.371429583587695836032807087447, −1.935472101595818822224280081007, 0.014001105417621767629151918761, 3.526714227184543689146295785946, 4.75663048434033938965301313099, 5.57482814465451561857889504578, 7.29746186663187685815110455059, 8.67489608376975122755631825544, 9.84598425337698162895725021083, 11.77016631189623617129559075149, 12.5070659667682363250424972010, 14.19867784085992649165321020302, 15.26412804954931839240674065011, 16.39222611395488261382299650023, 16.806492312951544759237361555798, 18.21911805501975771209321067559, 20.17398788587329450401426034724, 21.11876226276270615916127228463, 22.286395443780680080977404764141, 23.127268327916201789439376833727, 24.18727365556074731346939119281, 25.15657102173959804169074014746, 26.66395140509331634760083405414, 27.25100197873369830348981772447, 28.37696214000757825759147930374, 29.6761916142646840996891994435, 31.37327089186588329439347868547

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