Properties

Label 2-21e2-1.1-c3-0-19
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.95·2-s − 4.15·4-s + 11.8·5-s − 23.8·8-s + 23.3·10-s + 36.4·11-s + 0.964·13-s − 13.4·16-s + 98.2·17-s − 106.·19-s − 49.4·20-s + 71.3·22-s + 54.3·23-s + 16.3·25-s + 1.89·26-s + 229.·29-s + 127.·31-s + 164.·32-s + 192.·34-s + 311.·37-s − 207.·38-s − 283.·40-s − 419.·41-s + 523.·43-s − 151.·44-s + 106.·46-s + 270.·47-s + ⋯
L(s)  = 1  + 0.692·2-s − 0.519·4-s + 1.06·5-s − 1.05·8-s + 0.736·10-s + 0.998·11-s + 0.0205·13-s − 0.209·16-s + 1.40·17-s − 1.27·19-s − 0.552·20-s + 0.691·22-s + 0.492·23-s + 0.130·25-s + 0.0142·26-s + 1.47·29-s + 0.740·31-s + 0.907·32-s + 0.971·34-s + 1.38·37-s − 0.886·38-s − 1.11·40-s − 1.59·41-s + 1.85·43-s − 0.518·44-s + 0.341·46-s + 0.838·47-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: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.943530740\)
\(L(\frac12)\) \(\approx\) \(2.943530740\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.95T + 8T^{2} \)
5 \( 1 - 11.8T + 125T^{2} \)
11 \( 1 - 36.4T + 1.33e3T^{2} \)
13 \( 1 - 0.964T + 2.19e3T^{2} \)
17 \( 1 - 98.2T + 4.91e3T^{2} \)
19 \( 1 + 106.T + 6.85e3T^{2} \)
23 \( 1 - 54.3T + 1.21e4T^{2} \)
29 \( 1 - 229.T + 2.43e4T^{2} \)
31 \( 1 - 127.T + 2.97e4T^{2} \)
37 \( 1 - 311.T + 5.06e4T^{2} \)
41 \( 1 + 419.T + 6.89e4T^{2} \)
43 \( 1 - 523.T + 7.95e4T^{2} \)
47 \( 1 - 270.T + 1.03e5T^{2} \)
53 \( 1 + 251.T + 1.48e5T^{2} \)
59 \( 1 - 408.T + 2.05e5T^{2} \)
61 \( 1 + 860.T + 2.26e5T^{2} \)
67 \( 1 - 506.T + 3.00e5T^{2} \)
71 \( 1 + 523.T + 3.57e5T^{2} \)
73 \( 1 - 629.T + 3.89e5T^{2} \)
79 \( 1 - 319.T + 4.93e5T^{2} \)
83 \( 1 - 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + 348.T + 7.04e5T^{2} \)
97 \( 1 + 161.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58916478334278273655359448752, −9.724664404902149949101310473825, −9.069769103078658269714501535041, −8.088831695800062149855439549514, −6.51733542236772186648462893355, −5.95251475607399296641080653856, −4.90712013260357137549982833366, −3.91584836810128479261098429934, −2.65821569817968930762030531205, −1.06808365019296098869595337861, 1.06808365019296098869595337861, 2.65821569817968930762030531205, 3.91584836810128479261098429934, 4.90712013260357137549982833366, 5.95251475607399296641080653856, 6.51733542236772186648462893355, 8.088831695800062149855439549514, 9.069769103078658269714501535041, 9.724664404902149949101310473825, 10.58916478334278273655359448752

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