Properties

Label 2-221067-1.1-c1-0-26
Degree $2$
Conductor $221067$
Sign $-1$
Analytic cond. $1765.22$
Root an. cond. $42.0146$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s − 7-s + 4·16-s + 3·17-s + 2·19-s − 2·20-s + 6·23-s − 4·25-s + 2·28-s − 29-s − 8·31-s − 35-s + 2·37-s + 2·41-s + 43-s + 3·47-s + 49-s + 4·53-s − 3·59-s + 4·61-s − 8·64-s − 7·67-s − 6·68-s + 2·71-s + 2·73-s − 4·76-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.377·7-s + 16-s + 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s − 4/5·25-s + 0.377·28-s − 0.185·29-s − 1.43·31-s − 0.169·35-s + 0.328·37-s + 0.312·41-s + 0.152·43-s + 0.437·47-s + 1/7·49-s + 0.549·53-s − 0.390·59-s + 0.512·61-s − 64-s − 0.855·67-s − 0.727·68-s + 0.237·71-s + 0.234·73-s − 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221067\)    =    \(3^{2} \cdot 7 \cdot 11^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(1765.22\)
Root analytic conductor: \(42.0146\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{221067} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 221067,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
29 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.09191587244932, −12.86272299061095, −12.47895214036828, −11.78827096557052, −11.43044136543312, −10.71759797197585, −10.32808640922040, −9.868469665603888, −9.393175470961583, −9.007430189869462, −8.768620085785792, −7.836148324335340, −7.654961312259080, −7.110074633542056, −6.374806645593432, −5.906320765581436, −5.364420982264655, −5.125718788954090, −4.419632509133516, −3.769892983718794, −3.463005020154517, −2.809936465989325, −2.117945718983303, −1.334553401629509, −0.8011439841735829, 0, 0.8011439841735829, 1.334553401629509, 2.117945718983303, 2.809936465989325, 3.463005020154517, 3.769892983718794, 4.419632509133516, 5.125718788954090, 5.364420982264655, 5.906320765581436, 6.374806645593432, 7.110074633542056, 7.654961312259080, 7.836148324335340, 8.768620085785792, 9.007430189869462, 9.393175470961583, 9.868469665603888, 10.32808640922040, 10.71759797197585, 11.43044136543312, 11.78827096557052, 12.47895214036828, 12.86272299061095, 13.09191587244932

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