Properties

Degree 1
Conductor $ 3^{2} \cdot 7 $
Sign $-0.939 - 0.342i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + 8-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s − 17-s − 19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + 8-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s − 17-s − 19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.939 - 0.342i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.939 - 0.342i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(63\)    =    \(3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.939 - 0.342i$
motivic weight  =  \(0\)
character  :  $\chi_{63} (34, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 63,\ (1:\ ),\ -0.939 - 0.342i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1609175992 - 0.9126090552i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1609175992 - 0.9126090552i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6064060984 - 0.5088351335i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6064060984 - 0.5088351335i\)

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

−33.04505395566049150703762732496, −31.58553865087248007431878472696, −30.47149947748832369198563694883, −28.98637124911945001210240013104, −28.10699326839897084807315874411, −26.637322351001848101011857305252, −26.01493640650596700618066475321, −25.044141992410827421985443713078, −23.691220068777317727996842046732, −22.76492095257696082177378474635, −21.51076552900045323627886547966, −19.85463657287545624233967352120, −18.50654739504890602234359164559, −17.87451831229819133540196833690, −16.57381701716998256718886067077, −15.26660074020311617709482000335, −14.35784051823042400781620578528, −13.12793947599152688345099822989, −10.99621633307619684610401204736, −9.96031711711280314728317536698, −8.64894575264804548779482324867, −7.08118031881071500841259294816, −6.21908635095735866533630495706, −4.50636601233958095101447187123, −2.066485422968420720810869899729, 0.56929975421934199570280100575, 2.3056656336572927153605874930, 4.09945934075237740670677833466, 5.75753200078613424618053984761, 8.01548491746004374237530247642, 8.95271660885719401533689139493, 10.26400469809790399446654622119, 11.44895706453950623490206758788, 12.95087453247031168489827550594, 13.53927064785408013958605420191, 15.72649471839827997692803976964, 16.99876365587067851414758810634, 17.90876090382346868774729547969, 19.193747007396696851566620539335, 20.36596382861515845328735074954, 21.188605522800932291959603082685, 22.23633508935254910997470874787, 23.778619796475470875047435601147, 25.11636804269155075797828905573, 26.19258666981815687536306759032, 27.452585828236839116525043623220, 28.35107756861216251490525551465, 29.297341620179791481871767341083, 30.1914730423111638855293434465, 31.61924848461166507709057859695

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