Properties

Label 2-147-1.1-c3-0-10
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 3·3-s + 8·4-s + 4·5-s + 12·6-s + 9·9-s + 16·10-s + 62·11-s + 24·12-s + 62·13-s + 12·15-s − 64·16-s − 84·17-s + 36·18-s − 100·19-s + 32·20-s + 248·22-s − 42·23-s − 109·25-s + 248·26-s + 27·27-s − 10·29-s + 48·30-s + 48·31-s − 256·32-s + 186·33-s − 336·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.357·5-s + 0.816·6-s + 1/3·9-s + 0.505·10-s + 1.69·11-s + 0.577·12-s + 1.32·13-s + 0.206·15-s − 16-s − 1.19·17-s + 0.471·18-s − 1.20·19-s + 0.357·20-s + 2.40·22-s − 0.380·23-s − 0.871·25-s + 1.87·26-s + 0.192·27-s − 0.0640·29-s + 0.292·30-s + 0.278·31-s − 1.41·32-s + 0.981·33-s − 1.69·34-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: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.263859203\)
\(L(\frac12)\) \(\approx\) \(4.263859203\)
\(L(\frac{5}{2})\) 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 \( 1 - p^{2} T + p^{3} T^{2} \)
5 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 - 62 T + p^{3} T^{2} \)
13 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 + 42 T + p^{3} T^{2} \)
29 \( 1 + 10 T + p^{3} T^{2} \)
31 \( 1 - 48 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 - 248 T + p^{3} T^{2} \)
43 \( 1 - 68 T + p^{3} T^{2} \)
47 \( 1 + 324 T + p^{3} T^{2} \)
53 \( 1 - 258 T + p^{3} T^{2} \)
59 \( 1 + 120 T + p^{3} T^{2} \)
61 \( 1 + 622 T + p^{3} T^{2} \)
67 \( 1 - 904 T + p^{3} T^{2} \)
71 \( 1 + 678 T + p^{3} T^{2} \)
73 \( 1 - 642 T + p^{3} T^{2} \)
79 \( 1 - 740 T + p^{3} T^{2} \)
83 \( 1 + 468 T + p^{3} T^{2} \)
89 \( 1 + 200 T + p^{3} T^{2} \)
97 \( 1 - 1266 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90541366052443196022367010216, −11.85279352403554602686273887580, −10.90890111780282811089188845709, −9.316973031880648284907123329052, −8.556639843225078230326132189417, −6.67082009946559186424094448907, −6.06811797249981480709891027167, −4.36467015715741361044713715028, −3.64614525137488831061645082498, −1.95949944526276410048816982066, 1.95949944526276410048816982066, 3.64614525137488831061645082498, 4.36467015715741361044713715028, 6.06811797249981480709891027167, 6.67082009946559186424094448907, 8.556639843225078230326132189417, 9.316973031880648284907123329052, 10.90890111780282811089188845709, 11.85279352403554602686273887580, 12.90541366052443196022367010216

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