Properties

Degree $2$
Conductor $441$
Sign $1$
Motivic weight $3$
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: $1$
Motivic weight: \(3\)
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.322134996\)
\(L(\frac12)\) \(\approx\) \(1.322134996\)
\(L(\frac{5}{2})\) 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

−10.26503791358651125479145373839, −9.818079000207144636417257180115, −9.128608736344426805328728117576, −7.955211233003369077050306518078, −7.36580299173530388809600664016, −5.68216618093492650735470580051, −5.27772496209652220162957419395, −3.69913336318360698758648958574, −2.16981978401081169371352661618, −0.840820038756991393831167030671, 0.840820038756991393831167030671, 2.16981978401081169371352661618, 3.69913336318360698758648958574, 5.27772496209652220162957419395, 5.68216618093492650735470580051, 7.36580299173530388809600664016, 7.955211233003369077050306518078, 9.128608736344426805328728117576, 9.818079000207144636417257180115, 10.26503791358651125479145373839

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