Properties

Degree 1
Conductor $ 3^{2} \cdot 7 $
Sign $0.975 - 0.220i$
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)4-s − 5-s + 8-s + (0.5 + 0.866i)10-s + 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + 23-s + 25-s + (0.5 − 0.866i)26-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + 8-s + (0.5 + 0.866i)10-s + 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + 23-s + 25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.975 - 0.220i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.975 - 0.220i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(63\)    =    \(3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.975 - 0.220i$
motivic weight  =  \(0\)
character  :  $\chi_{63} (31, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 63,\ (1:\ ),\ 0.975 - 0.220i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.009068532 - 0.1126300311i$
$L(\frac12,\chi)$  $\approx$  $1.009068532 - 0.1126300311i$
$L(\chi,1)$  $\approx$  0.7721251257 - 0.1745403663i
$L(1,\chi)$  $\approx$  0.7721251257 - 0.1745403663i

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

−32.27234367388550506089539956400, −31.28834014941959263804163612217, −29.96758891475449991993986197236, −28.39582314837269400889392274829, −27.39011302290605991398353113837, −26.82796670935613978565861191717, −25.28726045518376063153938644955, −24.57128116719941385609681456635, −23.13020106327546438706788430997, −22.645981026903088723512508396852, −20.5249432869271806590231217374, −19.39429148102674447894349654389, −18.44498028951957130439477699101, −17.090190132324614371653558809257, −16.020463689615229918737142828277, −15.06239546133681930643429028589, −13.88891044009185368615955943932, −12.141960796503782292879499281509, −10.72877701622394508163292131764, −9.21788929630966320384634029624, −8.014996026395983674002173075677, −6.94083968489108383640188101493, −5.36675548432417486792891272818, −3.73331291513385510651575210250, −0.83327325022601813634542349622, 1.208219090186543188809201471520, 3.29993053263841558784128284112, 4.45056607310803967734795079924, 6.92713913626343929795325898470, 8.33530949887168361735404759706, 9.41140556084223642032906292863, 11.075908139655417464021079651530, 11.784747031979044447980676607, 13.066607361733385593164451821340, 14.62926047122161226357683094185, 16.223381223867594002329689963103, 17.262323244442977340349652932302, 18.78398123232665122131872388212, 19.46621528464890729997444958648, 20.54365964799939207609456838974, 21.80226343101008922835716030234, 22.88302897494825072961120702835, 24.11951405073786307700315063247, 25.75783157583616088467880602596, 26.76427788999370105949447346487, 27.76285269568187282247089220934, 28.51927555600703961791802260664, 29.95905032083069313855805417291, 30.76613434959428641946046209356, 31.663374751139834458388274007103

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