Properties

Label 2-21e2-1.1-c3-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.30·2-s + 20.1·4-s − 5.56·5-s − 64.6·8-s + 29.5·10-s + 13.9·11-s + 38.6·13-s + 181.·16-s − 43.4·17-s − 109.·19-s − 112.·20-s − 73.8·22-s + 74.8·23-s − 94.0·25-s − 205.·26-s + 72.3·29-s + 64.0·31-s − 447.·32-s + 230.·34-s + 188.·37-s + 578.·38-s + 359.·40-s + 24.7·41-s − 243.·43-s + 280.·44-s − 397.·46-s + 620.·47-s + ⋯
L(s)  = 1  − 1.87·2-s + 2.52·4-s − 0.497·5-s − 2.85·8-s + 0.933·10-s + 0.381·11-s + 0.825·13-s + 2.83·16-s − 0.620·17-s − 1.31·19-s − 1.25·20-s − 0.715·22-s + 0.678·23-s − 0.752·25-s − 1.54·26-s + 0.463·29-s + 0.371·31-s − 2.47·32-s + 1.16·34-s + 0.838·37-s + 2.47·38-s + 1.42·40-s + 0.0944·41-s − 0.864·43-s + 0.962·44-s − 1.27·46-s + 1.92·47-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: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.30T + 8T^{2} \)
5 \( 1 + 5.56T + 125T^{2} \)
11 \( 1 - 13.9T + 1.33e3T^{2} \)
13 \( 1 - 38.6T + 2.19e3T^{2} \)
17 \( 1 + 43.4T + 4.91e3T^{2} \)
19 \( 1 + 109.T + 6.85e3T^{2} \)
23 \( 1 - 74.8T + 1.21e4T^{2} \)
29 \( 1 - 72.3T + 2.43e4T^{2} \)
31 \( 1 - 64.0T + 2.97e4T^{2} \)
37 \( 1 - 188.T + 5.06e4T^{2} \)
41 \( 1 - 24.7T + 6.89e4T^{2} \)
43 \( 1 + 243.T + 7.95e4T^{2} \)
47 \( 1 - 620.T + 1.03e5T^{2} \)
53 \( 1 - 287.T + 1.48e5T^{2} \)
59 \( 1 + 525.T + 2.05e5T^{2} \)
61 \( 1 + 383.T + 2.26e5T^{2} \)
67 \( 1 - 198.T + 3.00e5T^{2} \)
71 \( 1 + 785.T + 3.57e5T^{2} \)
73 \( 1 + 331.T + 3.89e5T^{2} \)
79 \( 1 - 437.T + 4.93e5T^{2} \)
83 \( 1 + 241.T + 5.71e5T^{2} \)
89 \( 1 + 1.58e3T + 7.04e5T^{2} \)
97 \( 1 - 79.2T + 9.12e5T^{2} \)
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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

−10.22106376317108752559551344147, −9.120733074855839314550889579934, −8.596338009817352084355955264068, −7.77510770639747307369767302808, −6.80919370937447799556474738583, −6.05246425228804562535778906010, −4.12164881757758216710610186731, −2.61628169786358847898989233146, −1.30997429051290618949034423291, 0, 1.30997429051290618949034423291, 2.61628169786358847898989233146, 4.12164881757758216710610186731, 6.05246425228804562535778906010, 6.80919370937447799556474738583, 7.77510770639747307369767302808, 8.596338009817352084355955264068, 9.120733074855839314550889579934, 10.22106376317108752559551344147

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