Properties

Label 2-21e2-1.1-c3-0-39
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  + 5.27·2-s + 19.8·4-s + 10.5·5-s + 62.3·8-s + 55.6·10-s − 34.7·11-s + 37.2·13-s + 170.·16-s − 10.5·17-s + 58.5·19-s + 209.·20-s − 183.·22-s + 125.·23-s − 13.7·25-s + 196.·26-s + 35.4·29-s − 291.·31-s + 399.·32-s − 55.6·34-s − 259.·37-s + 309.·38-s + 658.·40-s − 338.·41-s + 6.80·43-s − 688.·44-s + 661.·46-s + 250.·47-s + ⋯
L(s)  = 1  + 1.86·2-s + 2.47·4-s + 0.943·5-s + 2.75·8-s + 1.75·10-s − 0.952·11-s + 0.795·13-s + 2.66·16-s − 0.150·17-s + 0.707·19-s + 2.33·20-s − 1.77·22-s + 1.13·23-s − 0.109·25-s + 1.48·26-s + 0.226·29-s − 1.69·31-s + 2.20·32-s − 0.280·34-s − 1.15·37-s + 1.31·38-s + 2.60·40-s − 1.28·41-s + 0.0241·43-s − 2.36·44-s + 2.11·46-s + 0.778·47-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

\(L(2)\) \(\approx\) \(7.225207247\)
\(L(\frac12)\) \(\approx\) \(7.225207247\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.27T + 8T^{2} \)
5 \( 1 - 10.5T + 125T^{2} \)
11 \( 1 + 34.7T + 1.33e3T^{2} \)
13 \( 1 - 37.2T + 2.19e3T^{2} \)
17 \( 1 + 10.5T + 4.91e3T^{2} \)
19 \( 1 - 58.5T + 6.85e3T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 - 35.4T + 2.43e4T^{2} \)
31 \( 1 + 291.T + 2.97e4T^{2} \)
37 \( 1 + 259.T + 5.06e4T^{2} \)
41 \( 1 + 338.T + 6.89e4T^{2} \)
43 \( 1 - 6.80T + 7.95e4T^{2} \)
47 \( 1 - 250.T + 1.03e5T^{2} \)
53 \( 1 - 536.T + 1.48e5T^{2} \)
59 \( 1 + 35.8T + 2.05e5T^{2} \)
61 \( 1 + 57.7T + 2.26e5T^{2} \)
67 \( 1 - 481.T + 3.00e5T^{2} \)
71 \( 1 + 363.T + 3.57e5T^{2} \)
73 \( 1 + 581.T + 3.89e5T^{2} \)
79 \( 1 + 693.T + 4.93e5T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + 353.T + 7.04e5T^{2} \)
97 \( 1 + 1.44e3T + 9.12e5T^{2} \)
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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

−10.90583753399711327422700441534, −10.24971021793301382508582500496, −8.900585609738174508222466442371, −7.46977265758075947592952363114, −6.62120739731328617301461187962, −5.53639256926580118373420190407, −5.20872362483547187682229648869, −3.80196629205402709802203622507, −2.80268322442694725187351413560, −1.68227702885123946317957318959, 1.68227702885123946317957318959, 2.80268322442694725187351413560, 3.80196629205402709802203622507, 5.20872362483547187682229648869, 5.53639256926580118373420190407, 6.62120739731328617301461187962, 7.46977265758075947592952363114, 8.900585609738174508222466442371, 10.24971021793301382508582500496, 10.90583753399711327422700441534

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