Properties

Degree 1
Conductor $ 3 \cdot 7 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s − 13-s + 16-s + 17-s − 19-s + 20-s + 22-s − 23-s + 25-s + 26-s − 29-s − 31-s − 32-s − 34-s + 37-s + 38-s − 40-s + 41-s + 43-s − 44-s + 46-s + 47-s + ⋯
L(s,χ)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s − 13-s + 16-s + 17-s − 19-s + 20-s + 22-s − 23-s + 25-s + 26-s − 29-s − 31-s − 32-s − 34-s + 37-s + 38-s − 40-s + 41-s + 43-s − 44-s + 46-s + 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{21} (20, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 21,\ (0:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.4972623804\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.4972623804\)
\(L(\chi,1)\)  \(\approx\)  \(0.6838072478\)
\(L(1,\chi)\)  \(\approx\)  \(0.6838072478\)

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

−39.50661155351630389171633460115, −38.21985593915273892734977795631, −36.87046584837538024852812190795, −36.286337297733217516955063033708, −34.455032738950987302447003018, −33.67499323055941205278829636710, −32.06905597513047654192922197302, −29.98408843411714656870834834775, −29.09345457094967826515164622089, −27.82592678096409045102785618249, −26.28746603707134008234353551778, −25.35180638669369017042649064483, −23.95175267810717684091926630693, −21.71236117436269543788775408707, −20.53983785320221716505090986872, −18.88189182634069320161874853493, −17.66389892894890481445748333359, −16.465154717935669189145654649009, −14.67082197914472783430852895830, −12.628728375934724403243638725586, −10.61054485467460396045181252678, −9.4646732073101131171217674859, −7.65463248516368468456673400912, −5.78036827357004952616082437239, −2.31518706430314115204629295971, 2.31518706430314115204629295971, 5.78036827357004952616082437239, 7.65463248516368468456673400912, 9.4646732073101131171217674859, 10.61054485467460396045181252678, 12.628728375934724403243638725586, 14.67082197914472783430852895830, 16.465154717935669189145654649009, 17.66389892894890481445748333359, 18.88189182634069320161874853493, 20.53983785320221716505090986872, 21.71236117436269543788775408707, 23.95175267810717684091926630693, 25.35180638669369017042649064483, 26.28746603707134008234353551778, 27.82592678096409045102785618249, 29.09345457094967826515164622089, 29.98408843411714656870834834775, 32.06905597513047654192922197302, 33.67499323055941205278829636710, 34.455032738950987302447003018, 36.286337297733217516955063033708, 36.87046584837538024852812190795, 38.21985593915273892734977795631, 39.50661155351630389171633460115

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