Properties

Label 2-21e2-1.1-c5-0-37
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  + 5.31·2-s − 3.71·4-s + 103.·5-s − 189.·8-s + 550.·10-s − 653.·11-s + 138.·13-s − 891.·16-s + 1.17e3·17-s + 1.71e3·19-s − 384.·20-s − 3.47e3·22-s + 4.02e3·23-s + 7.58e3·25-s + 734.·26-s − 2.64e3·29-s + 2.87e3·31-s + 1.33e3·32-s + 6.24e3·34-s + 2.85e3·37-s + 9.09e3·38-s − 1.96e4·40-s + 216.·41-s + 2.92e3·43-s + 2.42e3·44-s + 2.13e4·46-s + 1.48e4·47-s + ⋯
L(s)  = 1  + 0.940·2-s − 0.116·4-s + 1.85·5-s − 1.04·8-s + 1.74·10-s − 1.62·11-s + 0.226·13-s − 0.870·16-s + 0.985·17-s + 1.08·19-s − 0.215·20-s − 1.53·22-s + 1.58·23-s + 2.42·25-s + 0.212·26-s − 0.585·29-s + 0.537·31-s + 0.231·32-s + 0.926·34-s + 0.343·37-s + 1.02·38-s − 1.94·40-s + 0.0201·41-s + 0.241·43-s + 0.189·44-s + 1.48·46-s + 0.978·47-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

\(L(3)\) \(\approx\) \(4.294691190\)
\(L(\frac12)\) \(\approx\) \(4.294691190\)
\(L(\frac{7}{2})\) 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 \( 1 - 5.31T + 32T^{2} \)
5 \( 1 - 103.T + 3.12e3T^{2} \)
11 \( 1 + 653.T + 1.61e5T^{2} \)
13 \( 1 - 138.T + 3.71e5T^{2} \)
17 \( 1 - 1.17e3T + 1.41e6T^{2} \)
19 \( 1 - 1.71e3T + 2.47e6T^{2} \)
23 \( 1 - 4.02e3T + 6.43e6T^{2} \)
29 \( 1 + 2.64e3T + 2.05e7T^{2} \)
31 \( 1 - 2.87e3T + 2.86e7T^{2} \)
37 \( 1 - 2.85e3T + 6.93e7T^{2} \)
41 \( 1 - 216.T + 1.15e8T^{2} \)
43 \( 1 - 2.92e3T + 1.47e8T^{2} \)
47 \( 1 - 1.48e4T + 2.29e8T^{2} \)
53 \( 1 + 2.11e4T + 4.18e8T^{2} \)
59 \( 1 - 3.46e4T + 7.14e8T^{2} \)
61 \( 1 - 8.75e3T + 8.44e8T^{2} \)
67 \( 1 + 1.20e4T + 1.35e9T^{2} \)
71 \( 1 - 3.55e4T + 1.80e9T^{2} \)
73 \( 1 + 3.34e4T + 2.07e9T^{2} \)
79 \( 1 - 4.31e4T + 3.07e9T^{2} \)
83 \( 1 + 4.33e4T + 3.93e9T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 - 8.62e4T + 8.58e9T^{2} \)
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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

−10.17407961701712621996829745107, −9.586998262705515428248263924528, −8.686430263543575563575281851407, −7.36281059449699883745627711740, −6.09055580703777371985073619288, −5.40172447670850959456389475806, −4.95709707252150750463409143253, −3.21023858620297044783819171690, −2.47464217759359565939818430451, −0.967476634898867691799064602154, 0.967476634898867691799064602154, 2.47464217759359565939818430451, 3.21023858620297044783819171690, 4.95709707252150750463409143253, 5.40172447670850959456389475806, 6.09055580703777371985073619288, 7.36281059449699883745627711740, 8.686430263543575563575281851407, 9.586998262705515428248263924528, 10.17407961701712621996829745107

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