Properties

Label 2-21e2-1.1-c3-0-5
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.59·2-s − 5.44·4-s − 18.2·5-s − 21.4·8-s − 29.2·10-s − 61.2·11-s + 32.4·13-s + 9.23·16-s + 81.3·17-s − 20.9·19-s + 99.6·20-s − 97.9·22-s + 33.7·23-s + 209.·25-s + 51.8·26-s − 52.0·29-s + 193.·31-s + 186.·32-s + 129.·34-s − 267.·37-s − 33.4·38-s + 393.·40-s − 203.·41-s − 21.9·43-s + 333.·44-s + 53.9·46-s + 247.·47-s + ⋯
L(s)  = 1  + 0.564·2-s − 0.680·4-s − 1.63·5-s − 0.949·8-s − 0.924·10-s − 1.67·11-s + 0.692·13-s + 0.144·16-s + 1.16·17-s − 0.252·19-s + 1.11·20-s − 0.948·22-s + 0.305·23-s + 1.67·25-s + 0.391·26-s − 0.333·29-s + 1.12·31-s + 1.03·32-s + 0.655·34-s − 1.18·37-s − 0.142·38-s + 1.55·40-s − 0.773·41-s − 0.0778·43-s + 1.14·44-s + 0.172·46-s + 0.769·47-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: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9417593945\)
\(L(\frac12)\) \(\approx\) \(0.9417593945\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.59T + 8T^{2} \)
5 \( 1 + 18.2T + 125T^{2} \)
11 \( 1 + 61.2T + 1.33e3T^{2} \)
13 \( 1 - 32.4T + 2.19e3T^{2} \)
17 \( 1 - 81.3T + 4.91e3T^{2} \)
19 \( 1 + 20.9T + 6.85e3T^{2} \)
23 \( 1 - 33.7T + 1.21e4T^{2} \)
29 \( 1 + 52.0T + 2.43e4T^{2} \)
31 \( 1 - 193.T + 2.97e4T^{2} \)
37 \( 1 + 267.T + 5.06e4T^{2} \)
41 \( 1 + 203.T + 6.89e4T^{2} \)
43 \( 1 + 21.9T + 7.95e4T^{2} \)
47 \( 1 - 247.T + 1.03e5T^{2} \)
53 \( 1 + 140.T + 1.48e5T^{2} \)
59 \( 1 - 221.T + 2.05e5T^{2} \)
61 \( 1 - 652.T + 2.26e5T^{2} \)
67 \( 1 - 604.T + 3.00e5T^{2} \)
71 \( 1 + 716.T + 3.57e5T^{2} \)
73 \( 1 - 388.T + 3.89e5T^{2} \)
79 \( 1 + 289.T + 4.93e5T^{2} \)
83 \( 1 - 115.T + 5.71e5T^{2} \)
89 \( 1 + 939.T + 7.04e5T^{2} \)
97 \( 1 - 120.T + 9.12e5T^{2} \)
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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

−10.81573584444271084836590446217, −9.933168941347482001629587911642, −8.479008400130606201943177820739, −8.159703542587450712965418319666, −7.12998304207022536890470059098, −5.63425702340553180024569511130, −4.81875797720936963741585383860, −3.79444637563315078394793756403, −3.02051222280658579342969778407, −0.55675928207389041642844818240, 0.55675928207389041642844818240, 3.02051222280658579342969778407, 3.79444637563315078394793756403, 4.81875797720936963741585383860, 5.63425702340553180024569511130, 7.12998304207022536890470059098, 8.159703542587450712965418319666, 8.479008400130606201943177820739, 9.933168941347482001629587911642, 10.81573584444271084836590446217

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