Properties

Degree 1
Conductor $ 2^{3} \cdot 13 $
Sign $-0.252 + 0.967i$
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)3-s − 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)3-s − 5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)23-s + 25-s + 27-s + (0.5 + 0.866i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.252 + 0.967i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.252 + 0.967i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(104\)    =    \(2^{3} \cdot 13\)
\( \varepsilon \)  =  $-0.252 + 0.967i$
motivic weight  =  \(0\)
character  :  $\chi_{104} (35, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 104,\ (1:\ ),\ -0.252 + 0.967i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1063618709 + 0.1376971945i$
$L(\frac12,\chi)$  $\approx$  $0.1063618709 + 0.1376971945i$
$L(\chi,1)$  $\approx$  0.5940886554 - 0.1632753463i
$L(1,\chi)$  $\approx$  0.5940886554 - 0.1632753463i

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

−28.74277962505070557904353662098, −28.16347437125625367836539858567, −27.25010001348416179040472765632, −26.43599956693794714972648232950, −25.09494944675192453864232564857, −23.79988537780303966265905454619, −22.93693818255658800994487110139, −21.99611932897363211946083713411, −20.88496386042209911465839462536, −19.99674204320850640721330241432, −18.53977636868290620552973125956, −17.585905294877949674859230826701, −16.21410169830345824792633488197, −15.37795712036816480169719158816, −14.73132970085012457595790689263, −12.708383683079704550171050906225, −11.66012942230346853490518546841, −10.88427184403274438530649093259, −9.42364912718815163711893867176, −8.32493325215845953722271105448, −6.82139315423648894533401586957, −5.13450707579130745836325801173, −4.3733852460211915777991201951, −2.68486877521757785473401917745, −0.08411986133245054720994378796, 1.41468158242385770488805718284, 3.47783636632375730785943321104, 4.968454736790499123644963744174, 6.481171287991067464872883548908, 7.67913108913820139348248905774, 8.39009143360655193906214765384, 10.680873510828472431905033538473, 11.2774446497121714128940766110, 12.540492483571958258220223288489, 13.54067185358479051721915731383, 14.78471399209066712643112877622, 16.24913698009209615206643463339, 17.09461333488352122949314833217, 18.313242120109969796569205169637, 19.27323525478836914581763216062, 20.12082394744342097100259275990, 21.5538633748810307627838534318, 22.926295810406278825815474299442, 23.74619755161834181109286779765, 24.165597006987811729156304123496, 25.64766246130870855281154674904, 26.9484818413248893807801138148, 27.65109166206156463111336236075, 28.977370797293421143899313146875, 29.76912272160480054923618405036

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