Properties

Degree 1
Conductor $ 2^{2} \cdot 37 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s − 17-s + 19-s + 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s + 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s + 53-s + 55-s − 57-s + ⋯
L(s,χ)  = 1  − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s − 17-s + 19-s + 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s + 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s + 53-s + 55-s − 57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(148\)    =    \(2^{2} \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{148} (147, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 148,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5090392253$
$L(\frac12,\chi)$  $\approx$  $0.5090392253$
$L(\chi,1)$  $\approx$  0.5164746507
$L(1,\chi)$  $\approx$  0.5164746507

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

−28.055929388343378262423651035122, −26.86259990788137381775504418031, −26.33956052661445138027422280555, −24.62117613901229929362366241268, −23.9151107348469192225696089456, −22.72266633666223580699904520275, −22.48803894206718246379167085679, −21.07172436882948145435110476809, −19.759548636801391083349159962029, −18.94345085355237349573807131880, −17.90185163118126319447330078031, −16.68778427474795209157130932474, −15.88460037020232438567587937272, −15.16233624302250647841480000421, −13.25896422315704832833979452990, −12.47909993027605684188687266731, −11.47446809121786589720706013463, −10.4851387649294303925375890204, −9.33777644469170663419961742082, −7.6207396362520879513361003277, −6.840412086404635360259461624026, −5.422004887766601355090790590072, −4.331561162560183161429336269974, −2.84198705815088156092677036033, −0.50489835112590698642863946765, 0.50489835112590698642863946765, 2.84198705815088156092677036033, 4.331561162560183161429336269974, 5.422004887766601355090790590072, 6.840412086404635360259461624026, 7.6207396362520879513361003277, 9.33777644469170663419961742082, 10.4851387649294303925375890204, 11.47446809121786589720706013463, 12.47909993027605684188687266731, 13.25896422315704832833979452990, 15.16233624302250647841480000421, 15.88460037020232438567587937272, 16.68778427474795209157130932474, 17.90185163118126319447330078031, 18.94345085355237349573807131880, 19.759548636801391083349159962029, 21.07172436882948145435110476809, 22.48803894206718246379167085679, 22.72266633666223580699904520275, 23.9151107348469192225696089456, 24.62117613901229929362366241268, 26.33956052661445138027422280555, 26.86259990788137381775504418031, 28.055929388343378262423651035122

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