Properties

Label 2-21e2-1.1-c5-0-56
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $70.7292$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.88·2-s − 8.09·4-s + 42.7·5-s + 196.·8-s − 209.·10-s − 710.·11-s + 885.·13-s − 699.·16-s + 701.·17-s − 1.25e3·19-s − 345.·20-s + 3.47e3·22-s + 1.04e3·23-s − 1.29e3·25-s − 4.33e3·26-s − 6.15e3·29-s + 2.29e3·31-s − 2.85e3·32-s − 3.42e3·34-s + 404.·37-s + 6.14e3·38-s + 8.38e3·40-s + 1.78e4·41-s − 1.46e4·43-s + 5.75e3·44-s − 5.11e3·46-s + 2.11e4·47-s + ⋯
L(s)  = 1  − 0.864·2-s − 0.252·4-s + 0.764·5-s + 1.08·8-s − 0.661·10-s − 1.77·11-s + 1.45·13-s − 0.683·16-s + 0.588·17-s − 0.798·19-s − 0.193·20-s + 1.53·22-s + 0.412·23-s − 0.415·25-s − 1.25·26-s − 1.35·29-s + 0.428·31-s − 0.492·32-s − 0.508·34-s + 0.0485·37-s + 0.689·38-s + 0.828·40-s + 1.66·41-s − 1.20·43-s + 0.447·44-s − 0.356·46-s + 1.39·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(70.7292\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 4.88T + 32T^{2} \)
5 \( 1 - 42.7T + 3.12e3T^{2} \)
11 \( 1 + 710.T + 1.61e5T^{2} \)
13 \( 1 - 885.T + 3.71e5T^{2} \)
17 \( 1 - 701.T + 1.41e6T^{2} \)
19 \( 1 + 1.25e3T + 2.47e6T^{2} \)
23 \( 1 - 1.04e3T + 6.43e6T^{2} \)
29 \( 1 + 6.15e3T + 2.05e7T^{2} \)
31 \( 1 - 2.29e3T + 2.86e7T^{2} \)
37 \( 1 - 404.T + 6.93e7T^{2} \)
41 \( 1 - 1.78e4T + 1.15e8T^{2} \)
43 \( 1 + 1.46e4T + 1.47e8T^{2} \)
47 \( 1 - 2.11e4T + 2.29e8T^{2} \)
53 \( 1 + 107.T + 4.18e8T^{2} \)
59 \( 1 - 4.44e4T + 7.14e8T^{2} \)
61 \( 1 + 2.32e4T + 8.44e8T^{2} \)
67 \( 1 + 6.67e3T + 1.35e9T^{2} \)
71 \( 1 + 2.51e4T + 1.80e9T^{2} \)
73 \( 1 - 9.47e3T + 2.07e9T^{2} \)
79 \( 1 + 2.64e4T + 3.07e9T^{2} \)
83 \( 1 - 7.49e3T + 3.93e9T^{2} \)
89 \( 1 + 3.21e4T + 5.58e9T^{2} \)
97 \( 1 + 1.55e5T + 8.58e9T^{2} \)
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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

−9.880807297723089365377040109161, −8.951692828345099185930649171774, −8.193563719689487260459519985356, −7.41835151775526683993578082249, −5.99716703600493437055852760717, −5.24665984019617560045272701009, −3.93497501819272442383813250889, −2.43072112941586149287358492857, −1.25508954022680944596212275878, 0, 1.25508954022680944596212275878, 2.43072112941586149287358492857, 3.93497501819272442383813250889, 5.24665984019617560045272701009, 5.99716703600493437055852760717, 7.41835151775526683993578082249, 8.193563719689487260459519985356, 8.951692828345099185930649171774, 9.880807297723089365377040109161

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