Properties

Label 2-21e2-1.1-c5-0-35
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  + 0.790·2-s − 31.3·4-s + 104.·5-s − 50.1·8-s + 82.3·10-s + 497.·11-s − 206.·13-s + 964.·16-s − 63.1·17-s + 1.32e3·19-s − 3.26e3·20-s + 393.·22-s + 194.·23-s + 7.73e3·25-s − 163.·26-s − 4.32e3·29-s − 7.52e3·31-s + 2.36e3·32-s − 49.9·34-s + 1.03e4·37-s + 1.04e3·38-s − 5.22e3·40-s + 4.18e3·41-s + 5.96e3·43-s − 1.56e4·44-s + 153.·46-s + 4.38e3·47-s + ⋯
L(s)  = 1  + 0.139·2-s − 0.980·4-s + 1.86·5-s − 0.276·8-s + 0.260·10-s + 1.24·11-s − 0.338·13-s + 0.941·16-s − 0.0530·17-s + 0.841·19-s − 1.82·20-s + 0.173·22-s + 0.0766·23-s + 2.47·25-s − 0.0473·26-s − 0.954·29-s − 1.40·31-s + 0.408·32-s − 0.00740·34-s + 1.24·37-s + 0.117·38-s − 0.516·40-s + 0.388·41-s + 0.491·43-s − 1.21·44-s + 0.0107·46-s + 0.289·47-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: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.916595199\)
\(L(\frac12)\) \(\approx\) \(2.916595199\)
\(L(\frac{7}{2})\) 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 \( 1 - 0.790T + 32T^{2} \)
5 \( 1 - 104.T + 3.12e3T^{2} \)
11 \( 1 - 497.T + 1.61e5T^{2} \)
13 \( 1 + 206.T + 3.71e5T^{2} \)
17 \( 1 + 63.1T + 1.41e6T^{2} \)
19 \( 1 - 1.32e3T + 2.47e6T^{2} \)
23 \( 1 - 194.T + 6.43e6T^{2} \)
29 \( 1 + 4.32e3T + 2.05e7T^{2} \)
31 \( 1 + 7.52e3T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4T + 6.93e7T^{2} \)
41 \( 1 - 4.18e3T + 1.15e8T^{2} \)
43 \( 1 - 5.96e3T + 1.47e8T^{2} \)
47 \( 1 - 4.38e3T + 2.29e8T^{2} \)
53 \( 1 + 1.77e4T + 4.18e8T^{2} \)
59 \( 1 + 3.50e3T + 7.14e8T^{2} \)
61 \( 1 + 1.06e4T + 8.44e8T^{2} \)
67 \( 1 + 1.32e4T + 1.35e9T^{2} \)
71 \( 1 + 3.88e4T + 1.80e9T^{2} \)
73 \( 1 - 3.13e4T + 2.07e9T^{2} \)
79 \( 1 - 3.94e4T + 3.07e9T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 - 1.12e5T + 5.58e9T^{2} \)
97 \( 1 - 3.03e4T + 8.58e9T^{2} \)
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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

−9.999271517020814685596836761737, −9.261404847234857207941897346270, −9.087311572206446860905252177228, −7.51524579440770556153108667906, −6.26036482269676473445502944229, −5.61342783483642066867475689257, −4.67485491593308074048703653781, −3.40358415375551282883303464034, −1.98082051413090430184462553238, −0.926751260896687998395792421683, 0.926751260896687998395792421683, 1.98082051413090430184462553238, 3.40358415375551282883303464034, 4.67485491593308074048703653781, 5.61342783483642066867475689257, 6.26036482269676473445502944229, 7.51524579440770556153108667906, 9.087311572206446860905252177228, 9.261404847234857207941897346270, 9.999271517020814685596836761737

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