Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $-0.794 + 0.607i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.755 − 0.654i)2-s + (0.599 + 0.800i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.0713 − 0.997i)6-s + (−0.212 − 0.977i)7-s + (0.540 − 0.841i)8-s + (−0.281 + 0.959i)9-s + (0.281 + 0.959i)10-s + (0.540 + 0.841i)11-s + (−0.707 + 0.707i)12-s + (−0.479 + 0.877i)14-s + (−0.0713 − 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s,χ)  = 1  + (−0.755 − 0.654i)2-s + (0.599 + 0.800i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.0713 − 0.997i)6-s + (−0.212 − 0.977i)7-s + (0.540 − 0.841i)8-s + (−0.281 + 0.959i)9-s + (0.281 + 0.959i)10-s + (0.540 + 0.841i)11-s + (−0.707 + 0.707i)12-s + (−0.479 + 0.877i)14-s + (−0.0713 − 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.794 + 0.607i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.794 + 0.607i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $-0.794 + 0.607i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (333, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ -0.794 + 0.607i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.07733265676 + 0.2284675099i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.07733265676 + 0.2284675099i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6183090502 + 0.01060611886i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6183090502 + 0.01060611886i\)

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

−20.67925379735257699187577874186, −19.687789484387484306729039123673, −19.190420715954881691120939072377, −18.87963138310457566950091084893, −17.96910387633527143256639318409, −17.38022196261309658069288380317, −16.07733869097466723090758574642, −15.64728721473030845641779426969, −14.7612668805600488783419822889, −14.30263333208886117789160607564, −13.285859798477090048280081139622, −12.2827897550922631205673585349, −11.39850674184720668468671839151, −10.83325780110996998366359013182, −9.382658249535731485938204609799, −8.790158513495655546745597934590, −8.32494834192535284172284306304, −7.231893256238401914106857596745, −6.72678796865686189854501881190, −5.994873147261590069672828072418, −4.792973480037191564593402002210, −3.34316455457706691502186458611, −2.60721495401248239308819369216, −1.47694080336427749226576471875, −0.120368136734060614279701414423, 1.36718862311038943264858059140, 2.43592031640046773630052160892, 3.65803217053208267316623402935, 4.12437590526224972415864517309, 4.72772635691930475775929580882, 6.61342621295114528924112245594, 7.47111378549047488852462121238, 8.2924502979787700098001715009, 8.84341781708804806488851885109, 9.78654961082778217958836929861, 10.392435684583152415582199121, 11.16121189762903002585657094414, 11.991015356135379758816025575022, 12.97923089804889348129300485390, 13.52994423364207687381623126346, 14.928568963667022120956929406768, 15.346582415292429183303390789563, 16.458358774348193163363772184826, 16.9055150160512357633182888823, 17.48862897622472616743621532673, 19.02279345900107423072380748147, 19.343305397314527301921169668, 20.12389995284142404061903154446, 20.60380551287584823622382108574, 21.12359807941479638266153417044

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