Properties

Label 2-147-1.1-c3-0-15
Degree $2$
Conductor $147$
Sign $1$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.30·2-s + 3·3-s + 20.1·4-s − 5.56·5-s + 15.9·6-s + 64.6·8-s + 9·9-s − 29.5·10-s − 13.9·11-s + 60.5·12-s − 38.6·13-s − 16.6·15-s + 181.·16-s − 43.4·17-s + 47.7·18-s + 109.·19-s − 112.·20-s − 73.8·22-s − 74.8·23-s + 193.·24-s − 94.0·25-s − 205.·26-s + 27·27-s − 72.3·29-s − 88.5·30-s − 64.0·31-s + 447.·32-s + ⋯
L(s)  = 1  + 1.87·2-s + 0.577·3-s + 2.52·4-s − 0.497·5-s + 1.08·6-s + 2.85·8-s + 0.333·9-s − 0.933·10-s − 0.381·11-s + 1.45·12-s − 0.825·13-s − 0.287·15-s + 2.83·16-s − 0.620·17-s + 0.625·18-s + 1.31·19-s − 1.25·20-s − 0.715·22-s − 0.678·23-s + 1.64·24-s − 0.752·25-s − 1.54·26-s + 0.192·27-s − 0.463·29-s − 0.538·30-s − 0.371·31-s + 2.47·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.264506134\)
\(L(\frac12)\) \(\approx\) \(5.264506134\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
7 \( 1 \)
good2 \( 1 - 5.30T + 8T^{2} \)
5 \( 1 + 5.56T + 125T^{2} \)
11 \( 1 + 13.9T + 1.33e3T^{2} \)
13 \( 1 + 38.6T + 2.19e3T^{2} \)
17 \( 1 + 43.4T + 4.91e3T^{2} \)
19 \( 1 - 109.T + 6.85e3T^{2} \)
23 \( 1 + 74.8T + 1.21e4T^{2} \)
29 \( 1 + 72.3T + 2.43e4T^{2} \)
31 \( 1 + 64.0T + 2.97e4T^{2} \)
37 \( 1 - 188.T + 5.06e4T^{2} \)
41 \( 1 - 24.7T + 6.89e4T^{2} \)
43 \( 1 + 243.T + 7.95e4T^{2} \)
47 \( 1 - 620.T + 1.03e5T^{2} \)
53 \( 1 + 287.T + 1.48e5T^{2} \)
59 \( 1 + 525.T + 2.05e5T^{2} \)
61 \( 1 - 383.T + 2.26e5T^{2} \)
67 \( 1 - 198.T + 3.00e5T^{2} \)
71 \( 1 - 785.T + 3.57e5T^{2} \)
73 \( 1 - 331.T + 3.89e5T^{2} \)
79 \( 1 - 437.T + 4.93e5T^{2} \)
83 \( 1 + 241.T + 5.71e5T^{2} \)
89 \( 1 + 1.58e3T + 7.04e5T^{2} \)
97 \( 1 + 79.2T + 9.12e5T^{2} \)
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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

−12.73383853445324321625485449454, −11.93627522400671174494581971992, −11.05401242224449632042800787805, −9.706518151883140151115031150557, −7.87793926840287391271480386690, −7.09733109143947492928136762463, −5.69148201917935928372622410376, −4.55128028445737808601411205836, −3.48456369173179165373188270206, −2.26868367973133294118080355186, 2.26868367973133294118080355186, 3.48456369173179165373188270206, 4.55128028445737808601411205836, 5.69148201917935928372622410376, 7.09733109143947492928136762463, 7.87793926840287391271480386690, 9.706518151883140151115031150557, 11.05401242224449632042800787805, 11.93627522400671174494581971992, 12.73383853445324321625485449454

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