Properties

Label 2-21e2-1.1-c3-0-31
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  − 4.35·2-s + 11.0·4-s + 8.71·5-s − 13.0·8-s − 38.0·10-s + 43.5·11-s − 82·13-s − 30.9·16-s − 78.4·17-s + 20·19-s + 95.8·20-s − 190.·22-s + 130.·23-s − 48.9·25-s + 357.·26-s − 244.·29-s − 156·31-s + 239.·32-s + 342.·34-s + 186·37-s − 87.1·38-s − 114.·40-s − 165.·41-s + 164·43-s + 479.·44-s − 570·46-s + 470.·47-s + ⋯
L(s)  = 1  − 1.54·2-s + 1.37·4-s + 0.779·5-s − 0.577·8-s − 1.20·10-s + 1.19·11-s − 1.74·13-s − 0.484·16-s − 1.11·17-s + 0.241·19-s + 1.07·20-s − 1.84·22-s + 1.18·23-s − 0.391·25-s + 2.69·26-s − 1.56·29-s − 0.903·31-s + 1.32·32-s + 1.72·34-s + 0.826·37-s − 0.372·38-s − 0.450·40-s − 0.630·41-s + 0.581·43-s + 1.64·44-s − 1.82·46-s + 1.46·47-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 \( 1 + 4.35T + 8T^{2} \)
5 \( 1 - 8.71T + 125T^{2} \)
11 \( 1 - 43.5T + 1.33e3T^{2} \)
13 \( 1 + 82T + 2.19e3T^{2} \)
17 \( 1 + 78.4T + 4.91e3T^{2} \)
19 \( 1 - 20T + 6.85e3T^{2} \)
23 \( 1 - 130.T + 1.21e4T^{2} \)
29 \( 1 + 244.T + 2.43e4T^{2} \)
31 \( 1 + 156T + 2.97e4T^{2} \)
37 \( 1 - 186T + 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 - 470.T + 1.03e5T^{2} \)
53 \( 1 - 156.T + 1.48e5T^{2} \)
59 \( 1 - 156.T + 2.05e5T^{2} \)
61 \( 1 + 790T + 2.26e5T^{2} \)
67 \( 1 + 44T + 3.00e5T^{2} \)
71 \( 1 + 444.T + 3.57e5T^{2} \)
73 \( 1 + 126T + 3.89e5T^{2} \)
79 \( 1 + 712T + 4.93e5T^{2} \)
83 \( 1 + 1.46e3T + 5.71e5T^{2} \)
89 \( 1 + 1.45e3T + 7.04e5T^{2} \)
97 \( 1 + 798T + 9.12e5T^{2} \)
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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

−9.899663658060037205397210913282, −9.322126065354923390053054972524, −8.851801670679080149660290467972, −7.41647412442285102279247035085, −6.98883468530533569084981380967, −5.73427954776251340983509889552, −4.38070186325760608572410437539, −2.49829392819491818540720478451, −1.52048386907106017498380910051, 0, 1.52048386907106017498380910051, 2.49829392819491818540720478451, 4.38070186325760608572410437539, 5.73427954776251340983509889552, 6.98883468530533569084981380967, 7.41647412442285102279247035085, 8.851801670679080149660290467972, 9.322126065354923390053054972524, 9.899663658060037205397210913282

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