Properties

Label 2-21e2-1.1-c3-0-21
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  − 2.41·2-s − 2.17·4-s − 19.8·5-s + 24.5·8-s + 48.0·10-s − 23.9·11-s + 87.3·13-s − 41.9·16-s + 5.63·17-s + 64.8·19-s + 43.2·20-s + 57.7·22-s + 25.5·23-s + 270.·25-s − 210.·26-s − 60.3·29-s − 122.·31-s − 95.2·32-s − 13.6·34-s − 56.1·37-s − 156.·38-s − 488.·40-s + 299.·41-s − 501.·43-s + 51.9·44-s − 61.7·46-s − 305.·47-s + ⋯
L(s)  = 1  − 0.853·2-s − 0.271·4-s − 1.77·5-s + 1.08·8-s + 1.51·10-s − 0.656·11-s + 1.86·13-s − 0.654·16-s + 0.0804·17-s + 0.783·19-s + 0.483·20-s + 0.560·22-s + 0.232·23-s + 2.16·25-s − 1.59·26-s − 0.386·29-s − 0.710·31-s − 0.526·32-s − 0.0686·34-s − 0.249·37-s − 0.668·38-s − 1.93·40-s + 1.14·41-s − 1.77·43-s + 0.178·44-s − 0.198·46-s − 0.948·47-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 \( 1 + 2.41T + 8T^{2} \)
5 \( 1 + 19.8T + 125T^{2} \)
11 \( 1 + 23.9T + 1.33e3T^{2} \)
13 \( 1 - 87.3T + 2.19e3T^{2} \)
17 \( 1 - 5.63T + 4.91e3T^{2} \)
19 \( 1 - 64.8T + 6.85e3T^{2} \)
23 \( 1 - 25.5T + 1.21e4T^{2} \)
29 \( 1 + 60.3T + 2.43e4T^{2} \)
31 \( 1 + 122.T + 2.97e4T^{2} \)
37 \( 1 + 56.1T + 5.06e4T^{2} \)
41 \( 1 - 299.T + 6.89e4T^{2} \)
43 \( 1 + 501.T + 7.95e4T^{2} \)
47 \( 1 + 305.T + 1.03e5T^{2} \)
53 \( 1 - 375.T + 1.48e5T^{2} \)
59 \( 1 - 627.T + 2.05e5T^{2} \)
61 \( 1 - 3.75T + 2.26e5T^{2} \)
67 \( 1 + 813.T + 3.00e5T^{2} \)
71 \( 1 + 165.T + 3.57e5T^{2} \)
73 \( 1 - 619.T + 3.89e5T^{2} \)
79 \( 1 + 138.T + 4.93e5T^{2} \)
83 \( 1 + 621.T + 5.71e5T^{2} \)
89 \( 1 + 285.T + 7.04e5T^{2} \)
97 \( 1 + 603.T + 9.12e5T^{2} \)
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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

−10.33419811440724983581609669394, −9.088274030991434728730107760875, −8.357014619601616012426625719252, −7.82400630255851885196546771068, −6.93810883557655921831242610460, −5.33814560814580709424506664377, −4.14569041347904889351715367624, −3.37722073266509372629783289847, −1.16261129241548893594957761267, 0, 1.16261129241548893594957761267, 3.37722073266509372629783289847, 4.14569041347904889351715367624, 5.33814560814580709424506664377, 6.93810883557655921831242610460, 7.82400630255851885196546771068, 8.357014619601616012426625719252, 9.088274030991434728730107760875, 10.33419811440724983581609669394

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