Properties

Degree $2$
Conductor $441$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 7·13-s + 4·16-s − 7·19-s − 5·25-s − 7·31-s − 37-s + 5·43-s + 14·52-s + 14·61-s − 8·64-s + 11·67-s − 7·73-s + 14·76-s − 13·79-s + 14·97-s + 10·100-s − 7·103-s + 17·109-s + ⋯
L(s)  = 1  − 4-s − 1.94·13-s + 16-s − 1.60·19-s − 25-s − 1.25·31-s − 0.164·37-s + 0.762·43-s + 1.94·52-s + 1.79·61-s − 64-s + 1.34·67-s − 0.819·73-s + 1.60·76-s − 1.46·79-s + 1.42·97-s + 100-s − 0.689·103-s + 1.62·109-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−10.44340961493860792194432079598, −9.751017393898932451994893756079, −8.947372931910945992290137358883, −7.991372095607934979439494362982, −7.07511528958958291144252655588, −5.72591573809169746727814745787, −4.78354615730792598375845969221, −3.87851392965924754220515773132, −2.26817487861975221274698654515, 0, 2.26817487861975221274698654515, 3.87851392965924754220515773132, 4.78354615730792598375845969221, 5.72591573809169746727814745787, 7.07511528958958291144252655588, 7.991372095607934979439494362982, 8.947372931910945992290137358883, 9.751017393898932451994893756079, 10.44340961493860792194432079598

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