Properties

Degree 1
Conductor $ 5 \cdot 11 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 13-s + 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 23-s − 24-s + 26-s − 27-s + 28-s − 29-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯
L(s,χ)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 13-s + 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 23-s − 24-s + 26-s − 27-s + 28-s − 29-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{55} (54, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 55,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.431063069$
$L(\frac12,\chi)$  $\approx$  $2.431063069$
$L(\chi,1)$  $\approx$  1.694449067
$L(1,\chi)$  $\approx$  1.694449067

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

−32.992676266835819663823465156577, −31.802647200809233646907352040814, −30.36075907982639052006903792501, −29.86220196587676814080729191581, −28.393839274654861558429346263102, −27.61700479243725250676142644445, −25.796150909371293478307837457034, −24.38737830488568094702011150827, −23.60986162844672312553194126411, −22.68102708815235806998329686677, −21.41879925718685876264250429310, −20.71867097669620387938655234510, −18.851541393180054554974977118526, −17.44785743430645892914024382800, −16.30264279220135006390959232093, −15.108776587227819884423539999057, −13.77348440625003556944408026897, −12.40471443568932653648515943841, −11.40043623969198003422829975849, −10.42315246404025260876883624454, −7.92367740819724180850066627351, −6.37005874038062820681872530606, −5.25631708635296194324745449220, −3.97704030111212558605523918635, −1.59805935779877015404751613070, 1.59805935779877015404751613070, 3.97704030111212558605523918635, 5.25631708635296194324745449220, 6.37005874038062820681872530606, 7.92367740819724180850066627351, 10.42315246404025260876883624454, 11.40043623969198003422829975849, 12.40471443568932653648515943841, 13.77348440625003556944408026897, 15.108776587227819884423539999057, 16.30264279220135006390959232093, 17.44785743430645892914024382800, 18.851541393180054554974977118526, 20.71867097669620387938655234510, 21.41879925718685876264250429310, 22.68102708815235806998329686677, 23.60986162844672312553194126411, 24.38737830488568094702011150827, 25.796150909371293478307837457034, 27.61700479243725250676142644445, 28.393839274654861558429346263102, 29.86220196587676814080729191581, 30.36075907982639052006903792501, 31.802647200809233646907352040814, 32.992676266835819663823465156577

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