Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $0.986 - 0.162i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.858 − 0.512i)2-s + (0.473 + 0.880i)4-s + (−0.963 + 0.266i)5-s + (0.0448 − 0.998i)8-s + (0.963 + 0.266i)10-s + (−0.393 − 0.919i)11-s + (0.858 + 0.512i)13-s + (−0.550 + 0.834i)16-s + (−0.473 + 0.880i)17-s + (0.309 + 0.951i)19-s + (−0.691 − 0.722i)20-s + (−0.134 + 0.990i)22-s + (0.691 − 0.722i)23-s + (0.858 − 0.512i)25-s + (−0.473 − 0.880i)26-s + ⋯
L(s,χ)  = 1  + (−0.858 − 0.512i)2-s + (0.473 + 0.880i)4-s + (−0.963 + 0.266i)5-s + (0.0448 − 0.998i)8-s + (0.963 + 0.266i)10-s + (−0.393 − 0.919i)11-s + (0.858 + 0.512i)13-s + (−0.550 + 0.834i)16-s + (−0.473 + 0.880i)17-s + (0.309 + 0.951i)19-s + (−0.691 − 0.722i)20-s + (−0.134 + 0.990i)22-s + (0.691 − 0.722i)23-s + (0.858 − 0.512i)25-s + (−0.473 − 0.880i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.986 - 0.162i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.986 - 0.162i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.986 - 0.162i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (1070, \cdot )$
Sato-Tate  :  $\mu(70)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ 0.986 - 0.162i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.010658258 - 0.08255102754i$
$L(\frac12,\chi)$  $\approx$  $1.010658258 - 0.08255102754i$
$L(\chi,1)$  $\approx$  0.6785786993 - 0.08102751735i
$L(1,\chi)$  $\approx$  0.6785786993 - 0.08102751735i

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

−17.797906318478837724086401423994, −17.20912164878104203949153074237, −16.10810527584477717765900092837, −15.916619300032310649473771255209, −15.36834912001760118184559461704, −14.7954732028853447629822783180, −13.862697496273056547366110488318, −13.152744121616379431288133969905, −12.4005196278163940162176693458, −11.42228494949272146907280532751, −11.23637280085155723829534525490, −10.372958031959885906614729040877, −9.55528705868856573373934782835, −9.0630465302776759154822622736, −8.16546463027470363656788675528, −7.87741024088930549241131622446, −6.9084802411041141574044408317, −6.69853509087102877823786439887, −5.42557096584343337574124201626, −4.93688987666159393979766657851, −4.21342376778074302024025499734, −3.04697698772901214960328354369, −2.45128514127160285537472718607, −1.201486103856569036027882501638, −0.634043925170711020124840314, 0.661257275229703249916709576023, 1.33258732492721225509106007677, 2.50979515188262220433762497678, 3.044734885749419969390548832570, 3.97673877441123366673345822680, 4.239312127174357955387120393692, 5.6291436185513844051740029829, 6.48237687316447274693426434977, 6.96548983583561911356342161167, 8.027603681950972498999809114866, 8.32300123322621208690978092353, 8.797831162504521745282805280577, 9.84089680765517530234947055253, 10.514871712592184295802720460734, 11.104953728018220901010741653959, 11.49484803577143878581102111431, 12.27332666232380404184569754538, 12.90489040166745269958543217756, 13.630944067928155577252478623483, 14.49132830537589265184741170501, 15.30684328143770599076971281003, 15.94910992435771251712537305762, 16.38593798156593720080395704310, 16.99022835931221298762826605311, 17.87511399622590719674806674544

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