Properties

Degree $2$
Conductor $147$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 3-s − 1.82·4-s − 3.41·5-s − 0.414·6-s − 1.58·8-s + 9-s − 1.41·10-s − 2·11-s + 1.82·12-s − 2.58·13-s + 3.41·15-s + 3·16-s + 2.24·17-s + 0.414·18-s + 2.82·19-s + 6.24·20-s − 0.828·22-s − 7.65·23-s + 1.58·24-s + 6.65·25-s − 1.07·26-s − 27-s − 6.82·29-s + 1.41·30-s + 1.17·31-s + 4.41·32-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.577·3-s − 0.914·4-s − 1.52·5-s − 0.169·6-s − 0.560·8-s + 0.333·9-s − 0.447·10-s − 0.603·11-s + 0.527·12-s − 0.717·13-s + 0.881·15-s + 0.750·16-s + 0.543·17-s + 0.0976·18-s + 0.648·19-s + 1.39·20-s − 0.176·22-s − 1.59·23-s + 0.323·24-s + 1.33·25-s − 0.210·26-s − 0.192·27-s − 1.26·29-s + 0.258·30-s + 0.210·31-s + 0.780·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
good2 \( 1 - 0.414T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−12.24862634564251127101582445586, −11.95663633100051336715682572906, −10.62915661160969898296205034709, −9.527103915730750902096128604275, −8.134790574271130183505669457339, −7.42349770612555731170106759369, −5.65388414877058007429182791770, −4.57935404398144771969035979627, −3.53087914044105296455979885058, 0, 3.53087914044105296455979885058, 4.57935404398144771969035979627, 5.65388414877058007429182791770, 7.42349770612555731170106759369, 8.134790574271130183505669457339, 9.527103915730750902096128604275, 10.62915661160969898296205034709, 11.95663633100051336715682572906, 12.24862634564251127101582445586

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