↑

Properties

Label 1-21-21.20-r0-0-0
Degree $1$
Conductor $21$
Sign $1$
Analytic cond. $0.0975235$
Root an. cond. $0.0975235$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.0975235\)
Root analytic conductor: \(0.0975235\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 21,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4972623804\)
\(L(\frac12)\) \(\approx\) \(0.4972623804\)
\(L(1)\) \(\approx\) \(0.6838072478\)
\(L(1)\) \(\approx\) \(0.6838072478\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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