Properties

Label 2-147-21.20-c5-0-24
Degree $2$
Conductor $147$
Sign $-0.997 + 0.0660i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.80i·2-s + (12.4 + 9.40i)3-s − 14.2·4-s + 14.9·5-s + (−63.9 + 84.5i)6-s + 120. i·8-s + (66.1 + 233. i)9-s + 101. i·10-s + 96.0i·11-s + (−177. − 134. i)12-s − 416. i·13-s + (185. + 140. i)15-s − 1.27e3·16-s + 208.·17-s + (−1.59e3 + 449. i)18-s + 2.65e3i·19-s + ⋯
L(s)  = 1  + 1.20i·2-s + (0.797 + 0.603i)3-s − 0.446·4-s + 0.266·5-s + (−0.725 + 0.959i)6-s + 0.665i·8-s + (0.272 + 0.962i)9-s + 0.320i·10-s + 0.239i·11-s + (−0.356 − 0.269i)12-s − 0.684i·13-s + (0.212 + 0.160i)15-s − 1.24·16-s + 0.174·17-s + (−1.15 + 0.327i)18-s + 1.68i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0660i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.997 + 0.0660i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.997 + 0.0660i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.560280891\)
\(L(\frac12)\) \(\approx\) \(2.560280891\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-12.4 - 9.40i)T \)
7 \( 1 \)
good2 \( 1 - 6.80iT - 32T^{2} \)
5 \( 1 - 14.9T + 3.12e3T^{2} \)
11 \( 1 - 96.0iT - 1.61e5T^{2} \)
13 \( 1 + 416. iT - 3.71e5T^{2} \)
17 \( 1 - 208.T + 1.41e6T^{2} \)
19 \( 1 - 2.65e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.61e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.22e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.29e3iT - 2.86e7T^{2} \)
37 \( 1 - 3.67e3T + 6.93e7T^{2} \)
41 \( 1 - 1.54e4T + 1.15e8T^{2} \)
43 \( 1 + 6.01e3T + 1.47e8T^{2} \)
47 \( 1 + 3.90e3T + 2.29e8T^{2} \)
53 \( 1 + 3.62e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.82e3T + 7.14e8T^{2} \)
61 \( 1 + 2.01e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.39e4T + 1.35e9T^{2} \)
71 \( 1 + 2.60e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.09e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.01e4T + 3.07e9T^{2} \)
83 \( 1 - 1.07e5T + 3.93e9T^{2} \)
89 \( 1 - 1.36e5T + 5.58e9T^{2} \)
97 \( 1 - 2.51e4iT - 8.58e9T^{2} \)
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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

−13.06838580994504858881257297696, −11.59826233975886265436108578217, −10.26266726741721911572881046514, −9.442931795088519166551854069672, −8.095268881620763719018010167505, −7.69619898808192209162566534551, −6.13860611739010521477573916590, −5.17411278433439632910681584193, −3.70189196333010559546405790631, −2.08166107527363674109163709690, 0.76732301667370698493505831792, 2.06772490223051368575219469616, 2.99852156452375933983025759842, 4.33313168196755142010598566793, 6.37042336260037439220561299243, 7.37655309728457203900454839805, 8.870571161342025729943113782428, 9.520876706947542943967147242175, 10.75918068428201723543614187516, 11.67825788850553685113021252770

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