Properties

Label 2-21e2-1.1-c3-0-7
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 4·5-s + 16·10-s − 62·11-s + 62·13-s − 64·16-s + 84·17-s − 100·19-s − 32·20-s + 248·22-s + 42·23-s − 109·25-s − 248·26-s + 10·29-s + 48·31-s + 256·32-s − 336·34-s − 246·37-s + 400·38-s − 248·41-s + 68·43-s − 496·44-s − 168·46-s + 324·47-s + 436·50-s + 496·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.357·5-s + 0.505·10-s − 1.69·11-s + 1.32·13-s − 16-s + 1.19·17-s − 1.20·19-s − 0.357·20-s + 2.40·22-s + 0.380·23-s − 0.871·25-s − 1.87·26-s + 0.0640·29-s + 0.278·31-s + 1.41·32-s − 1.69·34-s − 1.09·37-s + 1.70·38-s − 0.944·41-s + 0.241·43-s − 1.69·44-s − 0.538·46-s + 1.00·47-s + 1.23·50-s + 1.32·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6182125143\)
\(L(\frac12)\) \(\approx\) \(0.6182125143\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + p^{2} T + p^{3} T^{2} \)
5 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 + 62 T + p^{3} T^{2} \)
13 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 - 84 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 - 42 T + p^{3} T^{2} \)
29 \( 1 - 10 T + p^{3} T^{2} \)
31 \( 1 - 48 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 + 248 T + p^{3} T^{2} \)
43 \( 1 - 68 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 + 258 T + p^{3} T^{2} \)
59 \( 1 - 120 T + p^{3} T^{2} \)
61 \( 1 + 622 T + p^{3} T^{2} \)
67 \( 1 - 904 T + p^{3} T^{2} \)
71 \( 1 - 678 T + p^{3} T^{2} \)
73 \( 1 - 642 T + p^{3} T^{2} \)
79 \( 1 - 740 T + p^{3} T^{2} \)
83 \( 1 - 468 T + p^{3} T^{2} \)
89 \( 1 - 200 T + p^{3} T^{2} \)
97 \( 1 - 1266 T + p^{3} T^{2} \)
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

−10.58756990037893644291082906655, −9.866244821808345581394716953170, −8.705784094021155244698261931591, −8.128861408024099967876178377529, −7.47016041190547561920798189508, −6.23858665548391049546352502219, −5.01662791105783350033959534517, −3.54261257552292440194803604581, −2.05382218505233693420147085435, −0.61177520299178034000328230038, 0.61177520299178034000328230038, 2.05382218505233693420147085435, 3.54261257552292440194803604581, 5.01662791105783350033959534517, 6.23858665548391049546352502219, 7.47016041190547561920798189508, 8.128861408024099967876178377529, 8.705784094021155244698261931591, 9.866244821808345581394716953170, 10.58756990037893644291082906655

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