Properties

Degree 1
Conductor $ 3^{2} \cdot 13 $
Sign $0.173 - 0.984i$
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 + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + 10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)14-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 + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + 10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)14-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.173 - 0.984i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.173 - 0.984i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(117\)    =    \(3^{2} \cdot 13\)
\( \varepsilon \)  =  $0.173 - 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{117} (103, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 117,\ (0:\ ),\ 0.173 - 0.984i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.067948710 - 0.8961153693i$
$L(\frac12,\chi)$  $\approx$  $1.067948710 - 0.8961153693i$
$L(\chi,1)$  $\approx$  1.191871764 - 0.6451893938i
$L(1,\chi)$  $\approx$  1.191871764 - 0.6451893938i

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

−29.63134872813522645666369492977, −27.99402309699696102059370805482, −27.67397571080857909159941415437, −25.923486151924814166776918340942, −25.24701670664513226296121133885, −24.47455002281503740062649296105, −23.52788457900165204566823163208, −22.34690275381276980624988073729, −21.33025928897640237727114435916, −20.63329698210662452313619923808, −18.92136894714334107780198520218, −17.56400491254571950646571853644, −17.056601077828851735643016379203, −15.72925792225618222449042885613, −14.84138873071722722790727567427, −13.75684615488050219548132008498, −12.55042380166292883903458349953, −11.882769792167515423037970636908, −9.712525819362819759116194508044, −8.7417779599640911777791877740, −7.676982780911402790092696873171, −6.10145036462221690319129487602, −5.224491509196370250724627779077, −4.073241575106952478982921086752, −2.054907577994694077965041526842, 1.43953985584737901792371024827, 2.98920976045348041017487324351, 4.129741553478762654431907276196, 5.65677282027855632809497717644, 6.836717704166833946768117426090, 8.58873242245059907122150638241, 10.14712450231741440092506827758, 10.73854477163926637754467237274, 11.82746536198846904289823290767, 13.247001496982654437303459483844, 14.22761675717029177979384549568, 14.73127870187947901489553337384, 16.615607600147492164747546577363, 17.842221201341040499068273627601, 18.82874890362890044480236480716, 19.75899135564110679668101418911, 21.00848195346114083790464876638, 21.67356231273020342641088518300, 22.76713059156623032799907290111, 23.59525726883986215426958047008, 24.72769415731251479498117731387, 26.19582288668527249596570850746, 27.09623126728098279482566981458, 28.02174522029504583508004906949, 29.499470504477703539537007746276

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