Properties

Label 2-21e2-1.1-c3-0-23
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 $1$

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: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−10.11475228142303876399599195621, −9.227059925730514287466817924483, −8.456655412017598751789836830545, −7.62831977428277707497391886396, −6.91444832273275243292503653981, −5.25078624997037342326830925106, −4.12511813104430031466636897007, −3.52696004030330274073052593649, −1.23583670498177209247174643099, 0, 1.23583670498177209247174643099, 3.52696004030330274073052593649, 4.12511813104430031466636897007, 5.25078624997037342326830925106, 6.91444832273275243292503653981, 7.62831977428277707497391886396, 8.456655412017598751789836830545, 9.227059925730514287466817924483, 10.11475228142303876399599195621

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