Properties

Label 2-21e2-1.1-c5-0-13
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  + 3.09·2-s − 22.4·4-s − 13.7·5-s − 168.·8-s − 42.6·10-s + 3.66·11-s − 780.·13-s + 197.·16-s − 50.0·17-s + 1.06e3·19-s + 309.·20-s + 11.3·22-s − 4.10e3·23-s − 2.93e3·25-s − 2.41e3·26-s + 1.48e3·29-s + 5.51e3·31-s + 5.99e3·32-s − 154.·34-s + 6.14e3·37-s + 3.28e3·38-s + 2.32e3·40-s − 1.07e4·41-s + 1.76e4·43-s − 82.2·44-s − 1.26e4·46-s − 2.94e4·47-s + ⋯
L(s)  = 1  + 0.546·2-s − 0.701·4-s − 0.246·5-s − 0.929·8-s − 0.134·10-s + 0.00913·11-s − 1.28·13-s + 0.193·16-s − 0.0420·17-s + 0.675·19-s + 0.173·20-s + 0.00499·22-s − 1.61·23-s − 0.939·25-s − 0.699·26-s + 0.328·29-s + 1.03·31-s + 1.03·32-s − 0.0229·34-s + 0.737·37-s + 0.369·38-s + 0.229·40-s − 0.999·41-s + 1.45·43-s − 0.00640·44-s − 0.883·46-s − 1.94·47-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

\(L(3)\) \(\approx\) \(1.372175755\)
\(L(\frac12)\) \(\approx\) \(1.372175755\)
\(L(\frac{7}{2})\) 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 \( 1 - 3.09T + 32T^{2} \)
5 \( 1 + 13.7T + 3.12e3T^{2} \)
11 \( 1 - 3.66T + 1.61e5T^{2} \)
13 \( 1 + 780.T + 3.71e5T^{2} \)
17 \( 1 + 50.0T + 1.41e6T^{2} \)
19 \( 1 - 1.06e3T + 2.47e6T^{2} \)
23 \( 1 + 4.10e3T + 6.43e6T^{2} \)
29 \( 1 - 1.48e3T + 2.05e7T^{2} \)
31 \( 1 - 5.51e3T + 2.86e7T^{2} \)
37 \( 1 - 6.14e3T + 6.93e7T^{2} \)
41 \( 1 + 1.07e4T + 1.15e8T^{2} \)
43 \( 1 - 1.76e4T + 1.47e8T^{2} \)
47 \( 1 + 2.94e4T + 2.29e8T^{2} \)
53 \( 1 - 1.92e4T + 4.18e8T^{2} \)
59 \( 1 + 6.61e3T + 7.14e8T^{2} \)
61 \( 1 - 3.67e4T + 8.44e8T^{2} \)
67 \( 1 - 4.69e4T + 1.35e9T^{2} \)
71 \( 1 - 4.16e4T + 1.80e9T^{2} \)
73 \( 1 - 2.94e4T + 2.07e9T^{2} \)
79 \( 1 - 2.21e4T + 3.07e9T^{2} \)
83 \( 1 - 3.89e3T + 3.93e9T^{2} \)
89 \( 1 + 2.05e4T + 5.58e9T^{2} \)
97 \( 1 - 1.77e4T + 8.58e9T^{2} \)
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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

−9.968051464714525039216989960339, −9.689325973371482403044040384685, −8.392466406323971086083290089306, −7.67693881452740139581803628545, −6.40025371599518669071548806812, −5.37330380138153902068624709081, −4.51350319740305998498649929404, −3.58650631837099791146819413697, −2.33245959779544026687990431913, −0.54328570081352305606004310810, 0.54328570081352305606004310810, 2.33245959779544026687990431913, 3.58650631837099791146819413697, 4.51350319740305998498649929404, 5.37330380138153902068624709081, 6.40025371599518669071548806812, 7.67693881452740139581803628545, 8.392466406323971086083290089306, 9.689325973371482403044040384685, 9.968051464714525039216989960339

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