Properties

Degree 1
Conductor $ 2^{3} \cdot 3^{2} $
Sign $0.766 + 0.642i$
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)5-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 35-s − 37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 35-s − 37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.766 + 0.642i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.766 + 0.642i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(72\)    =    \(2^{3} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.766 + 0.642i$
motivic weight  =  \(0\)
character  :  $\chi_{72} (13, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 72,\ (0:\ ),\ 0.766 + 0.642i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9197214877 + 0.3347512453i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9197214877 + 0.3347512453i\)
\(L(\chi,1)\)  \(\approx\)  \(1.035205736 + 0.2134755561i\)
\(L(1,\chi)\)  \(\approx\)  \(1.035205736 + 0.2134755561i\)

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

−31.71905557787174441538596254243, −30.17375131833142524894927598414, −29.48421580941850672573034809243, −28.163561540372382389635507260242, −27.42542855836235172091111798517, −25.72186739323419272356986733161, −25.26132519103447050704413751036, −23.72435340757309339707721143610, −22.9122346915152802393354639874, −21.46661557659550218001629314767, −20.36519722341295997147191642723, −19.60071893007075376862556111168, −17.85631056047438788033707824402, −16.983241051665548410315856266625, −15.937349668853705008737854827317, −14.39184439348052417964155640728, −13.1595509650851485054737479288, −12.3133561253029776154034714594, −10.484692531373141296415547819509, −9.52242381613333263134807422902, −8.08358848300956800406786806195, −6.57991586841707826170401745110, −5.10629851236419492999352337039, −3.629394625628101399085476972088, −1.42467481873239634771439473912, 2.22597710360548271714684502500, 3.669186280593918975584787366677, 5.82255136630110305200079307601, 6.6234689328207247566273298344, 8.491082575023549961928390227699, 9.69775569278157298222034893172, 11.007121727582652084083529766350, 12.23350348113595704231288338606, 13.73846673582274412699715473714, 14.67400468122114456720768227004, 16.00173978074129690032718703065, 17.2041748756147054666435420123, 18.81551373666136698933800628125, 18.98622330756815119322015081036, 21.01379150198451652054836439490, 21.85934278670287855152163372819, 22.780363239488029427159042612252, 24.17102377076190764350303422985, 25.44082151383310473372851304468, 26.10979805331162340313045100797, 27.41517636242089709720143388095, 28.6139255685274555331863775509, 29.619248052843133548409247498817, 30.557319028048212582972783100226, 31.81345037823403048832936559512

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