Properties

Label 2-21e2-1.1-c5-0-61
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  + 8.71·2-s + 44.0·4-s + 99.6·5-s + 104.·8-s + 868.·10-s + 374.·11-s + 868.·13-s − 495.·16-s − 1.09e3·17-s − 868.·19-s + 4.38e3·20-s + 3.26e3·22-s + 2.97e3·23-s + 6.81e3·25-s + 7.57e3·26-s + 4.51e3·29-s − 9.55e3·31-s − 7.67e3·32-s − 9.55e3·34-s − 5.46e3·37-s − 7.57e3·38-s + 1.04e4·40-s + 9.86e3·41-s + 1.25e4·43-s + 1.64e4·44-s + 2.59e4·46-s − 9.96e3·47-s + ⋯
L(s)  = 1  + 1.54·2-s + 1.37·4-s + 1.78·5-s + 0.577·8-s + 2.74·10-s + 0.934·11-s + 1.42·13-s − 0.484·16-s − 0.920·17-s − 0.552·19-s + 2.45·20-s + 1.43·22-s + 1.17·23-s + 2.17·25-s + 2.19·26-s + 0.997·29-s − 1.78·31-s − 1.32·32-s − 1.41·34-s − 0.656·37-s − 0.851·38-s + 1.03·40-s + 0.916·41-s + 1.03·43-s + 1.28·44-s + 1.80·46-s − 0.658·47-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

\(L(3)\) \(\approx\) \(8.032512331\)
\(L(\frac12)\) \(\approx\) \(8.032512331\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.71T + 32T^{2} \)
5 \( 1 - 99.6T + 3.12e3T^{2} \)
11 \( 1 - 374.T + 1.61e5T^{2} \)
13 \( 1 - 868.T + 3.71e5T^{2} \)
17 \( 1 + 1.09e3T + 1.41e6T^{2} \)
19 \( 1 + 868.T + 2.47e6T^{2} \)
23 \( 1 - 2.97e3T + 6.43e6T^{2} \)
29 \( 1 - 4.51e3T + 2.05e7T^{2} \)
31 \( 1 + 9.55e3T + 2.86e7T^{2} \)
37 \( 1 + 5.46e3T + 6.93e7T^{2} \)
41 \( 1 - 9.86e3T + 1.15e8T^{2} \)
43 \( 1 - 1.25e4T + 1.47e8T^{2} \)
47 \( 1 + 9.96e3T + 2.29e8T^{2} \)
53 \( 1 + 1.51e4T + 4.18e8T^{2} \)
59 \( 1 - 4.26e4T + 7.14e8T^{2} \)
61 \( 1 + 4.77e4T + 8.44e8T^{2} \)
67 \( 1 + 2.99e4T + 1.35e9T^{2} \)
71 \( 1 - 6.14e4T + 1.80e9T^{2} \)
73 \( 1 + 4.86e4T + 2.07e9T^{2} \)
79 \( 1 - 8.01e4T + 3.07e9T^{2} \)
83 \( 1 - 3.07e4T + 3.93e9T^{2} \)
89 \( 1 + 2.08e4T + 5.58e9T^{2} \)
97 \( 1 + 1.33e5T + 8.58e9T^{2} \)
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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

−10.70082444135937252094094836590, −9.260190183982918571532015724875, −8.854998182208820853177482868653, −6.82587963752568901565701215656, −6.29512335163101504674777760737, −5.60087651655389706781391121257, −4.60144330627534084647586392951, −3.50545723784076952286868138194, −2.33924826763764349662170238856, −1.35622425856082421627766480521, 1.35622425856082421627766480521, 2.33924826763764349662170238856, 3.50545723784076952286868138194, 4.60144330627534084647586392951, 5.60087651655389706781391121257, 6.29512335163101504674777760737, 6.82587963752568901565701215656, 8.854998182208820853177482868653, 9.260190183982918571532015724875, 10.70082444135937252094094836590

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