Properties

Degree 1
Conductor $ 2^{3} \cdot 127 $
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 − 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s + 53-s − 55-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 − 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s + 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1016\)    =    \(2^{3} \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1016} (253, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1016,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.253924112$
$L(\frac12,\chi)$  $\approx$  $3.253924112$
$L(\chi,1)$  $\approx$  1.576968438
$L(1,\chi)$  $\approx$  1.576968438

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

−21.25268286503159065291065915081, −20.83015178093231663366035065935, −19.73027985886277748336545084921, −19.206069508757536642599765796007, −18.449442312018656577891270795770, −17.56674527490329537448861824173, −16.66397675715987317123789057248, −15.82821361137024094531375485637, −15.09380646939201568405400529756, −14.05376740125206652368517299513, −13.73627178102414991365637806567, −12.55965662131605612717761310999, −12.45660558807179424992551889288, −10.46030921063116868207070500039, −10.11489129865145873889970328277, −9.4446156714605272902815197354, −8.51283892299492366080687750596, −7.6371565968330169914344907618, −6.73119235162422725411456044377, −5.841389240415433389151076876739, −4.8302493695832161705020716109, −3.70409332165738140536478928533, −2.567353241348533321313266159994, −2.319428013873288434997152834098, −0.77476751040949914418414486720, 0.77476751040949914418414486720, 2.319428013873288434997152834098, 2.567353241348533321313266159994, 3.70409332165738140536478928533, 4.8302493695832161705020716109, 5.841389240415433389151076876739, 6.73119235162422725411456044377, 7.6371565968330169914344907618, 8.51283892299492366080687750596, 9.4446156714605272902815197354, 10.11489129865145873889970328277, 10.46030921063116868207070500039, 12.45660558807179424992551889288, 12.55965662131605612717761310999, 13.73627178102414991365637806567, 14.05376740125206652368517299513, 15.09380646939201568405400529756, 15.82821361137024094531375485637, 16.66397675715987317123789057248, 17.56674527490329537448861824173, 18.449442312018656577891270795770, 19.206069508757536642599765796007, 19.73027985886277748336545084921, 20.83015178093231663366035065935, 21.25268286503159065291065915081

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