Properties

Label 2-21e2-1.1-c3-0-8
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  − 2.27·2-s − 2.82·4-s − 4.54·5-s + 24.6·8-s + 10.3·10-s + 40.7·11-s − 53.2·13-s − 33.4·16-s + 4.54·17-s − 122.·19-s + 12.8·20-s − 92.7·22-s − 131.·23-s − 104.·25-s + 121.·26-s + 216.·29-s + 251.·31-s − 120.·32-s − 10.3·34-s + 11.8·37-s + 278.·38-s − 112.·40-s − 111.·41-s + 369.·43-s − 115.·44-s + 298.·46-s − 262.·47-s + ⋯
L(s)  = 1  − 0.804·2-s − 0.353·4-s − 0.406·5-s + 1.08·8-s + 0.327·10-s + 1.11·11-s − 1.13·13-s − 0.522·16-s + 0.0649·17-s − 1.48·19-s + 0.143·20-s − 0.898·22-s − 1.19·23-s − 0.834·25-s + 0.914·26-s + 1.38·29-s + 1.45·31-s − 0.668·32-s − 0.0522·34-s + 0.0528·37-s + 1.19·38-s − 0.442·40-s − 0.425·41-s + 1.30·43-s − 0.394·44-s + 0.957·46-s − 0.815·47-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

\(L(2)\) \(\approx\) \(0.7807484555\)
\(L(\frac12)\) \(\approx\) \(0.7807484555\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.27T + 8T^{2} \)
5 \( 1 + 4.54T + 125T^{2} \)
11 \( 1 - 40.7T + 1.33e3T^{2} \)
13 \( 1 + 53.2T + 2.19e3T^{2} \)
17 \( 1 - 4.54T + 4.91e3T^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 + 131.T + 1.21e4T^{2} \)
29 \( 1 - 216.T + 2.43e4T^{2} \)
31 \( 1 - 251.T + 2.97e4T^{2} \)
37 \( 1 - 11.8T + 5.06e4T^{2} \)
41 \( 1 + 111.T + 6.89e4T^{2} \)
43 \( 1 - 369.T + 7.95e4T^{2} \)
47 \( 1 + 262.T + 1.03e5T^{2} \)
53 \( 1 - 567.T + 1.48e5T^{2} \)
59 \( 1 - 839.T + 2.05e5T^{2} \)
61 \( 1 - 485.T + 2.26e5T^{2} \)
67 \( 1 + 333.T + 3.00e5T^{2} \)
71 \( 1 + 590.T + 3.57e5T^{2} \)
73 \( 1 + 490.T + 3.89e5T^{2} \)
79 \( 1 - 121.T + 4.93e5T^{2} \)
83 \( 1 - 609.T + 5.71e5T^{2} \)
89 \( 1 - 719.T + 7.04e5T^{2} \)
97 \( 1 - 637.T + 9.12e5T^{2} \)
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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

−10.29312676949778676866035382490, −9.902982592825101480440539840035, −8.790806704786590083336837896943, −8.199470125953334476160793993620, −7.21673092787523816716364967762, −6.19164934299687274337274066824, −4.65342779442294263022442699013, −3.98311374031340365598012413412, −2.16558080544106776519264121809, −0.63506450749297929860961787280, 0.63506450749297929860961787280, 2.16558080544106776519264121809, 3.98311374031340365598012413412, 4.65342779442294263022442699013, 6.19164934299687274337274066824, 7.21673092787523816716364967762, 8.199470125953334476160793993620, 8.790806704786590083336837896943, 9.902982592825101480440539840035, 10.29312676949778676866035382490

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