Properties

Label 2-21e2-1.1-c3-0-11
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·4-s + 7·5-s + 24·8-s − 14·10-s + 5·11-s + 14·13-s − 16·16-s − 21·17-s − 49·19-s − 28·20-s − 10·22-s + 159·23-s − 76·25-s − 28·26-s − 58·29-s − 147·31-s − 160·32-s + 42·34-s + 219·37-s + 98·38-s + 168·40-s + 350·41-s − 124·43-s − 20·44-s − 318·46-s + 525·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.626·5-s + 1.06·8-s − 0.442·10-s + 0.137·11-s + 0.298·13-s − 1/4·16-s − 0.299·17-s − 0.591·19-s − 0.313·20-s − 0.0969·22-s + 1.44·23-s − 0.607·25-s − 0.211·26-s − 0.371·29-s − 0.851·31-s − 0.883·32-s + 0.211·34-s + 0.973·37-s + 0.418·38-s + 0.664·40-s + 1.33·41-s − 0.439·43-s − 0.0685·44-s − 1.01·46-s + 1.62·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.217379995\)
\(L(\frac12)\) \(\approx\) \(1.217379995\)
\(L(\frac{5}{2})\) 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 \( 1 + p T + p^{3} T^{2} \)
5 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 - 5 T + p^{3} T^{2} \)
13 \( 1 - 14 T + p^{3} T^{2} \)
17 \( 1 + 21 T + p^{3} T^{2} \)
19 \( 1 + 49 T + p^{3} T^{2} \)
23 \( 1 - 159 T + p^{3} T^{2} \)
29 \( 1 + 2 p T + p^{3} T^{2} \)
31 \( 1 + 147 T + p^{3} T^{2} \)
37 \( 1 - 219 T + p^{3} T^{2} \)
41 \( 1 - 350 T + p^{3} T^{2} \)
43 \( 1 + 124 T + p^{3} T^{2} \)
47 \( 1 - 525 T + p^{3} T^{2} \)
53 \( 1 + 303 T + p^{3} T^{2} \)
59 \( 1 + 105 T + p^{3} T^{2} \)
61 \( 1 - 413 T + p^{3} T^{2} \)
67 \( 1 - 415 T + p^{3} T^{2} \)
71 \( 1 - 432 T + p^{3} T^{2} \)
73 \( 1 - 1113 T + p^{3} T^{2} \)
79 \( 1 + 103 T + p^{3} T^{2} \)
83 \( 1 - 1092 T + p^{3} T^{2} \)
89 \( 1 + 329 T + p^{3} T^{2} \)
97 \( 1 - 882 T + p^{3} T^{2} \)
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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

−10.62985483352710740329405706014, −9.525747041259070586450859722476, −9.125446613062804473119740580057, −8.140759859595263919306718204977, −7.16849850080254152034593932205, −6.00989367231886593468631191056, −4.94486694299807537213281936660, −3.82236014641521100135577302809, −2.14752756553276567996908627563, −0.806389519578029808987152562838, 0.806389519578029808987152562838, 2.14752756553276567996908627563, 3.82236014641521100135577302809, 4.94486694299807537213281936660, 6.00989367231886593468631191056, 7.16849850080254152034593932205, 8.140759859595263919306718204977, 9.125446613062804473119740580057, 9.525747041259070586450859722476, 10.62985483352710740329405706014

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