Properties

Degree 1
Conductor $ 2^{2} \cdot 13 $
Sign $0.711 - 0.702i$
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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.711 - 0.702i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.711 - 0.702i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(52\)    =    \(2^{2} \cdot 13\)
\( \varepsilon \)  =  $0.711 - 0.702i$
motivic weight  =  \(0\)
character  :  $\chi_{52} (3, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 52,\ (1:\ ),\ 0.711 - 0.702i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.982770984 - 0.8138532600i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.982770984 - 0.8138532600i\)
\(L(\chi,1)\)  \(\approx\)  \(1.460222993 - 0.3812477201i\)
\(L(1,\chi)\)  \(\approx\)  \(1.460222993 - 0.3812477201i\)

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

−33.07442184972861923944137476912, −32.64062339061599163542143235238, −31.00412380606351041606015914060, −30.09085514473229209698581535361, −28.64699601266026094374315772330, −27.55560941361184879854647722606, −26.34698299040620397252368215864, −25.576188736632627808992540384180, −24.29080798967969853806392301574, −22.65487788603621474569935750955, −21.57930938719149185461982082023, −20.59843655100138817988906801396, −19.65405032570482338954403786318, −17.65868265594300846058723622672, −16.939488056099957288882923853069, −15.2981237317562762265909484423, −14.23248852074343061922338650243, −13.21626707717390909814638059477, −11.09740926757498083893157884053, −9.98909510841826610155436269656, −8.945998351174364585446495123163, −7.198092654673264343997172931972, −5.273494634605773076838741649855, −3.92125738045100425198357315497, −1.9520464967871546869327975157, 1.49822208265687061109279131252, 2.9059847727421980431917766389, 5.45592640868744703473300792598, 6.68090083472023769975174485460, 8.40895081334319114935450569619, 9.36737334305044522412781719937, 11.345369969190425916671907977624, 12.668769108959763130106505335383, 13.8933173921017815599084530820, 14.73571689647835057769478847139, 16.63835464195437891585540718849, 18.09593883107125966084389820669, 18.648597134081762969207911539483, 20.23039383070662110173127043504, 21.34242550617004668704884155074, 22.54993655599350479127027143738, 24.39172655741410356227758086134, 24.78142970527927812691140697738, 25.89532386794198401258343807136, 27.28069928844560711590175100291, 28.902220984479481328630339790239, 29.55025386832094499887010616903, 30.80804044061694231996438982436, 31.730202550868152344861877412811, 32.990703130516555897941466339846

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