Properties

Label 2-21e2-1.1-c5-0-30
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.29·2-s − 3.99·4-s + 37.0·5-s − 190.·8-s + 196.·10-s − 227.·11-s + 518·13-s − 880.·16-s − 777.·17-s + 1.48e3·19-s − 148.·20-s − 1.20e3·22-s + 3.82e3·23-s − 1.75e3·25-s + 2.74e3·26-s + 105.·29-s + 2.60e3·31-s + 1.43e3·32-s − 4.11e3·34-s + 402·37-s + 7.85e3·38-s − 7.05e3·40-s + 629.·41-s + 6.95e3·43-s + 910.·44-s + 2.02e4·46-s + 2.73e4·47-s + ⋯
L(s)  = 1  + 0.935·2-s − 0.124·4-s + 0.662·5-s − 1.05·8-s + 0.619·10-s − 0.566·11-s + 0.850·13-s − 0.859·16-s − 0.652·17-s + 0.943·19-s − 0.0828·20-s − 0.530·22-s + 1.50·23-s − 0.560·25-s + 0.795·26-s + 0.0233·29-s + 0.486·31-s + 0.248·32-s − 0.610·34-s + 0.0482·37-s + 0.882·38-s − 0.697·40-s + 0.0585·41-s + 0.573·43-s + 0.0708·44-s + 1.41·46-s + 1.80·47-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: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.338381495\)
\(L(\frac12)\) \(\approx\) \(3.338381495\)
\(L(\frac{7}{2})\) 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 \( 1 - 5.29T + 32T^{2} \)
5 \( 1 - 37.0T + 3.12e3T^{2} \)
11 \( 1 + 227.T + 1.61e5T^{2} \)
13 \( 1 - 518T + 3.71e5T^{2} \)
17 \( 1 + 777.T + 1.41e6T^{2} \)
19 \( 1 - 1.48e3T + 2.47e6T^{2} \)
23 \( 1 - 3.82e3T + 6.43e6T^{2} \)
29 \( 1 - 105.T + 2.05e7T^{2} \)
31 \( 1 - 2.60e3T + 2.86e7T^{2} \)
37 \( 1 - 402T + 6.93e7T^{2} \)
41 \( 1 - 629.T + 1.15e8T^{2} \)
43 \( 1 - 6.95e3T + 1.47e8T^{2} \)
47 \( 1 - 2.73e4T + 2.29e8T^{2} \)
53 \( 1 - 3.05e4T + 4.18e8T^{2} \)
59 \( 1 + 4.51e4T + 7.14e8T^{2} \)
61 \( 1 - 2.26e4T + 8.44e8T^{2} \)
67 \( 1 + 1.31e4T + 1.35e9T^{2} \)
71 \( 1 + 4.82e4T + 1.80e9T^{2} \)
73 \( 1 - 8.28e4T + 2.07e9T^{2} \)
79 \( 1 + 8.11e4T + 3.07e9T^{2} \)
83 \( 1 - 6.66e4T + 3.93e9T^{2} \)
89 \( 1 - 1.26e5T + 5.58e9T^{2} \)
97 \( 1 - 1.06e4T + 8.58e9T^{2} \)
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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

−10.39551768231754969497378276588, −9.313869612851715149417540206959, −8.709727463876749284332781287888, −7.38462437858985662654650672347, −6.20430282197908445374540718509, −5.50439842443360897298497574752, −4.60488513476385655306588856344, −3.46734384938181125147657633571, −2.43441286083528667871770078338, −0.834942799138849304109410216631, 0.834942799138849304109410216631, 2.43441286083528667871770078338, 3.46734384938181125147657633571, 4.60488513476385655306588856344, 5.50439842443360897298497574752, 6.20430282197908445374540718509, 7.38462437858985662654650672347, 8.709727463876749284332781287888, 9.313869612851715149417540206959, 10.39551768231754969497378276588

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