Properties

Degree 1
Conductor $ 3^{2} \cdot 7 $
Sign $0.592 - 0.805i$
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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.592 - 0.805i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.592 - 0.805i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(63\)    =    \(3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.592 - 0.805i$
motivic weight  =  \(0\)
character  :  $\chi_{63} (4, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 63,\ (0:\ ),\ 0.592 - 0.805i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7171710840 - 0.3625137677i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7171710840 - 0.3625137677i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8379244875 - 0.3133759179i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8379244875 - 0.3133759179i\)

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

−32.864832066351855367368332985728, −31.798845492420927951044902269706, −30.23412667191290870177421505015, −28.9486766450324522839998949958, −28.070610696555864509588425229046, −26.73400642638878563934405877734, −25.85007082781836539138985861782, −24.81536634570934709820180691451, −23.98325751917814622810078118516, −22.48503253242194658991599267489, −21.48877610368606330640471113593, −19.768641098694839253719897504213, −18.71626068736256780297147619079, −17.27243260516527457390545165920, −16.92421683909599246546521774541, −15.19129758523393687024668901429, −14.22216859070808330312693840317, −13.07323471740285680643896143714, −11.058270689532387661158297818593, −9.602714660579852966276029314009, −8.83267428611754168144913616075, −7.00587111139890202004336352453, −6.05663292553839477074012870504, −4.521434042363181263721744792389, −1.815323927137525955850154845701, 1.61823011729816480489603275011, 3.179660044695541330267763160784, 5.04482459327279130369001975293, 6.92102361092146309450821706977, 8.686461192307386427843332142167, 9.708514515951531548888379021801, 10.80594822326727780372506464341, 12.24070244105722268587363133470, 13.33698052823308211865679015300, 14.586083039714685039462603669877, 16.63194994413362820895584510725, 17.50272124872854959471415760018, 18.51753856081684662710620633265, 19.8402150986012437466029997415, 20.80918457439383955825463555863, 21.94790699879680793319694978701, 22.74486303834643485784364066382, 24.820194214566040103329617573296, 25.540276702119093919862647776817, 26.968206068168425981117027885885, 27.756415087443877904083552110674, 29.21268334996917744745905048485, 29.60431010899603816874775582863, 30.82910415226818429590032029509, 32.09134217965152011768646938980

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