Properties

Label 2-147-1.1-c3-0-12
Degree $2$
Conductor $147$
Sign $-1$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 7·4-s + 12·5-s + 3·6-s + 15·8-s + 9·9-s − 12·10-s + 20·11-s + 21·12-s − 84·13-s − 36·15-s + 41·16-s − 96·17-s − 9·18-s + 12·19-s − 84·20-s − 20·22-s − 176·23-s − 45·24-s + 19·25-s + 84·26-s − 27·27-s + 58·29-s + 36·30-s − 264·31-s − 161·32-s + ⋯
L(s)  = 1  − 0.353·2-s − 0.577·3-s − 7/8·4-s + 1.07·5-s + 0.204·6-s + 0.662·8-s + 1/3·9-s − 0.379·10-s + 0.548·11-s + 0.505·12-s − 1.79·13-s − 0.619·15-s + 0.640·16-s − 1.36·17-s − 0.117·18-s + 0.144·19-s − 0.939·20-s − 0.193·22-s − 1.59·23-s − 0.382·24-s + 0.151·25-s + 0.633·26-s − 0.192·27-s + 0.371·29-s + 0.219·30-s − 1.52·31-s − 0.889·32-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: yes
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -1)\)

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 \( 1 + p T \)
7 \( 1 \)
good2 \( 1 + T + p^{3} T^{2} \)
5 \( 1 - 12 T + p^{3} T^{2} \)
11 \( 1 - 20 T + p^{3} T^{2} \)
13 \( 1 + 84 T + p^{3} T^{2} \)
17 \( 1 + 96 T + p^{3} T^{2} \)
19 \( 1 - 12 T + p^{3} T^{2} \)
23 \( 1 + 176 T + p^{3} T^{2} \)
29 \( 1 - 2 p T + p^{3} T^{2} \)
31 \( 1 + 264 T + p^{3} T^{2} \)
37 \( 1 - 258 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 - 156 T + p^{3} T^{2} \)
47 \( 1 + 408 T + p^{3} T^{2} \)
53 \( 1 + 722 T + p^{3} T^{2} \)
59 \( 1 - 492 T + p^{3} T^{2} \)
61 \( 1 + 492 T + p^{3} T^{2} \)
67 \( 1 - 412 T + p^{3} T^{2} \)
71 \( 1 - 296 T + p^{3} T^{2} \)
73 \( 1 - 240 T + p^{3} T^{2} \)
79 \( 1 - 776 T + p^{3} T^{2} \)
83 \( 1 - 924 T + p^{3} T^{2} \)
89 \( 1 + 744 T + p^{3} T^{2} \)
97 \( 1 + 168 T + p^{3} T^{2} \)
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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

−12.20195726756650422241073522087, −10.88007903490593513500051988762, −9.672135827053330853225560735685, −9.454514758526488831897198280441, −7.903931117775059840010464044635, −6.56029385957782858486388932900, −5.35999822990028879429235403923, −4.32594732984773623637445150922, −1.98366565515813513033987205203, 0, 1.98366565515813513033987205203, 4.32594732984773623637445150922, 5.35999822990028879429235403923, 6.56029385957782858486388932900, 7.903931117775059840010464044635, 9.454514758526488831897198280441, 9.672135827053330853225560735685, 10.88007903490593513500051988762, 12.20195726756650422241073522087

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