Properties

Degree $2$
Conductor $147$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 3-s + 3.82·4-s + 0.585·5-s − 2.41·6-s − 4.41·8-s + 9-s − 1.41·10-s − 2·11-s + 3.82·12-s + 5.41·13-s + 0.585·15-s + 2.99·16-s + 6.24·17-s − 2.41·18-s + 2.82·19-s + 2.24·20-s + 4.82·22-s + 3.65·23-s − 4.41·24-s − 4.65·25-s − 13.0·26-s + 27-s − 1.17·29-s − 1.41·30-s − 6.82·31-s + 1.58·32-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.91·4-s + 0.261·5-s − 0.985·6-s − 1.56·8-s + 0.333·9-s − 0.447·10-s − 0.603·11-s + 1.10·12-s + 1.50·13-s + 0.151·15-s + 0.749·16-s + 1.51·17-s − 0.569·18-s + 0.648·19-s + 0.501·20-s + 1.02·22-s + 0.762·23-s − 0.901·24-s − 0.931·25-s − 2.56·26-s + 0.192·27-s − 0.217·29-s − 0.258·30-s − 1.22·31-s + 0.280·32-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: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682448\)
\(L(\frac12)\) \(\approx\) \(0.682448\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 - 0.585T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 - 6.24T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 - 3.75T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 5.89T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 - 5.41T + 97T^{2} \)
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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

−13.06252264540380797503225997297, −11.64815720039895623060251437167, −10.63626188559369251023440765868, −9.827557758340378217934002094453, −8.913542592088079760457981218470, −8.052294528463012176197637921293, −7.19369438058058265519089458762, −5.71762903499348270420993315024, −3.29104047009395636747641337910, −1.51610409940336560074145371519, 1.51610409940336560074145371519, 3.29104047009395636747641337910, 5.71762903499348270420993315024, 7.19369438058058265519089458762, 8.052294528463012176197637921293, 8.913542592088079760457981218470, 9.827557758340378217934002094453, 10.63626188559369251023440765868, 11.64815720039895623060251437167, 13.06252264540380797503225997297

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