Properties

Label 2-221067-1.1-c1-0-36
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 $2$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·5-s + 7-s − 4·13-s + 4·16-s − 17-s − 7·19-s − 4·20-s − 25-s − 2·28-s − 29-s − 3·31-s + 2·35-s + 8·37-s − 6·41-s − 12·43-s − 8·47-s + 49-s + 8·52-s + 6·53-s − 8·59-s − 13·61-s − 8·64-s − 8·65-s + 67-s + 2·68-s − 11·73-s + ⋯
L(s)  = 1  − 4-s + 0.894·5-s + 0.377·7-s − 1.10·13-s + 16-s − 0.242·17-s − 1.60·19-s − 0.894·20-s − 1/5·25-s − 0.377·28-s − 0.185·29-s − 0.538·31-s + 0.338·35-s + 1.31·37-s − 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 1.10·52-s + 0.824·53-s − 1.04·59-s − 1.66·61-s − 64-s − 0.992·65-s + 0.122·67-s + 0.242·68-s − 1.28·73-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: \(2\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−13.40223571191151, −13.06897888589063, −12.66857775620220, −12.06134440479932, −11.69755764723322, −10.99726344588289, −10.49645462966850, −10.07791628626079, −9.726510120074051, −9.221894062371248, −8.820206037545403, −8.292303755486947, −7.874540399583287, −7.326532993176729, −6.647219653719628, −6.178804111785584, −5.722228274663432, −5.086898916834527, −4.715901063155706, −4.345533961858258, −3.650215560732966, −3.014190070607411, −2.306545177533922, −1.815920846156140, −1.267601101525550, 0, 0, 1.267601101525550, 1.815920846156140, 2.306545177533922, 3.014190070607411, 3.650215560732966, 4.345533961858258, 4.715901063155706, 5.086898916834527, 5.722228274663432, 6.178804111785584, 6.647219653719628, 7.326532993176729, 7.874540399583287, 8.292303755486947, 8.820206037545403, 9.221894062371248, 9.726510120074051, 10.07791628626079, 10.49645462966850, 10.99726344588289, 11.69755764723322, 12.06134440479932, 12.66857775620220, 13.06897888589063, 13.40223571191151

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