Properties

Label 2-21e2-1.1-c5-0-2
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  − 3.38·2-s − 20.5·4-s − 54.5·5-s + 177.·8-s + 184.·10-s − 481.·11-s − 512.·13-s + 57.6·16-s + 590.·17-s − 2.45e3·19-s + 1.12e3·20-s + 1.62e3·22-s − 1.77e3·23-s − 151.·25-s + 1.73e3·26-s + 4.24e3·29-s − 9.76e3·31-s − 5.88e3·32-s − 1.99e3·34-s − 9.96e3·37-s + 8.28e3·38-s − 9.68e3·40-s − 3.37e3·41-s − 1.82e4·43-s + 9.91e3·44-s + 5.99e3·46-s + 1.32e3·47-s + ⋯
L(s)  = 1  − 0.597·2-s − 0.642·4-s − 0.975·5-s + 0.981·8-s + 0.582·10-s − 1.20·11-s − 0.841·13-s + 0.0562·16-s + 0.495·17-s − 1.55·19-s + 0.627·20-s + 0.717·22-s − 0.699·23-s − 0.0486·25-s + 0.502·26-s + 0.937·29-s − 1.82·31-s − 1.01·32-s − 0.296·34-s − 1.19·37-s + 0.930·38-s − 0.957·40-s − 0.313·41-s − 1.50·43-s + 0.772·44-s + 0.418·46-s + 0.0872·47-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

\(L(3)\) \(\approx\) \(0.06529418052\)
\(L(\frac12)\) \(\approx\) \(0.06529418052\)
\(L(\frac{7}{2})\) 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 \( 1 + 3.38T + 32T^{2} \)
5 \( 1 + 54.5T + 3.12e3T^{2} \)
11 \( 1 + 481.T + 1.61e5T^{2} \)
13 \( 1 + 512.T + 3.71e5T^{2} \)
17 \( 1 - 590.T + 1.41e6T^{2} \)
19 \( 1 + 2.45e3T + 2.47e6T^{2} \)
23 \( 1 + 1.77e3T + 6.43e6T^{2} \)
29 \( 1 - 4.24e3T + 2.05e7T^{2} \)
31 \( 1 + 9.76e3T + 2.86e7T^{2} \)
37 \( 1 + 9.96e3T + 6.93e7T^{2} \)
41 \( 1 + 3.37e3T + 1.15e8T^{2} \)
43 \( 1 + 1.82e4T + 1.47e8T^{2} \)
47 \( 1 - 1.32e3T + 2.29e8T^{2} \)
53 \( 1 + 3.48e4T + 4.18e8T^{2} \)
59 \( 1 - 1.15e4T + 7.14e8T^{2} \)
61 \( 1 + 3.14e4T + 8.44e8T^{2} \)
67 \( 1 - 2.75e4T + 1.35e9T^{2} \)
71 \( 1 - 2.28e4T + 1.80e9T^{2} \)
73 \( 1 + 1.59e4T + 2.07e9T^{2} \)
79 \( 1 - 8.71e4T + 3.07e9T^{2} \)
83 \( 1 - 9.03e4T + 3.93e9T^{2} \)
89 \( 1 + 1.26e5T + 5.58e9T^{2} \)
97 \( 1 - 1.65e4T + 8.58e9T^{2} \)
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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

−10.32001036510897946719819448375, −9.405477807014397945130404011145, −8.233643609730644569906427311798, −7.967699637826928587522133733213, −6.91701222726895263995424564101, −5.35365465137933013451861377593, −4.51691690914004799435291349173, −3.46732193142051158906078907578, −1.93277724139956044442823774617, −0.13960430164166760584063765954, 0.13960430164166760584063765954, 1.93277724139956044442823774617, 3.46732193142051158906078907578, 4.51691690914004799435291349173, 5.35365465137933013451861377593, 6.91701222726895263995424564101, 7.967699637826928587522133733213, 8.233643609730644569906427311798, 9.405477807014397945130404011145, 10.32001036510897946719819448375

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