↑

Properties

Label 2-147-21.20-c5-0-7
Degree $2$
Conductor $147$
Sign $0.990 + 0.140i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 10.7i·2-s + (14.9 − 4.31i)3-s − 83.7·4-s − 26.3·5-s + (−46.3 − 161. i)6-s + 556. i·8-s + (205. − 129. i)9-s + 283. i·10-s + 604. i·11-s + (−1.25e3 + 360. i)12-s + 732. i·13-s + (−394. + 113. i)15-s + 3.30e3·16-s − 1.69e3·17-s + (−1.38e3 − 2.21e3i)18-s + 211. i·19-s + ⋯
L(s)  = 1  − 1.90i·2-s + (0.961 − 0.276i)3-s − 2.61·4-s − 0.470·5-s + (−0.525 − 1.82i)6-s + 3.07i·8-s + (0.847 − 0.531i)9-s + 0.895i·10-s + 1.50i·11-s + (−2.51 + 0.723i)12-s + 1.20i·13-s + (−0.452 + 0.130i)15-s + 3.23·16-s − 1.42·17-s + (−1.01 − 1.61i)18-s + 0.134i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.990 + 0.140i$
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: \((2,\ 147,\ (\ :5/2),\ 0.990 + 0.140i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9505174854\)
\(L(\frac12)\) \(\approx\) \(0.9505174854\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-14.9 + 4.31i)T \)
7 \( 1 \)
good2 \( 1 + 10.7iT - 32T^{2} \)
5 \( 1 + 26.3T + 3.12e3T^{2} \)
11 \( 1 - 604. iT - 1.61e5T^{2} \)
13 \( 1 - 732. iT - 3.71e5T^{2} \)
17 \( 1 + 1.69e3T + 1.41e6T^{2} \)
19 \( 1 - 211. iT - 2.47e6T^{2} \)
23 \( 1 - 966. iT - 6.43e6T^{2} \)
29 \( 1 + 764. iT - 2.05e7T^{2} \)
31 \( 1 - 79.9iT - 2.86e7T^{2} \)
37 \( 1 + 643.T + 6.93e7T^{2} \)
41 \( 1 + 5.20e3T + 1.15e8T^{2} \)
43 \( 1 - 5.84e3T + 1.47e8T^{2} \)
47 \( 1 + 1.07e4T + 2.29e8T^{2} \)
53 \( 1 + 9.76e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.04e4T + 7.14e8T^{2} \)
61 \( 1 - 4.71e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.85e4T + 1.35e9T^{2} \)
71 \( 1 + 2.88e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.29e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.08e4T + 3.07e9T^{2} \)
83 \( 1 + 3.39e4T + 3.93e9T^{2} \)
89 \( 1 + 8.43e4T + 5.58e9T^{2} \)
97 \( 1 - 1.70e4iT - 8.58e9T^{2} \)
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

−12.07517126738917747195786580095, −11.31309057789278046886502712092, −9.988200060608703378124425176234, −9.344685582894624663700389243770, −8.418540885704184202317079548085, −7.08179694545780658620232156800, −4.53447143454719746989583404350, −3.91522298771949276933450344844, −2.40593224169182205848931155397, −1.62915834103039341951130428785, 0.28874719074153822762844119646, 3.33232527865356693476229287893, 4.49699864298053191569154675724, 5.75961102184206077159668949606, 6.95925441382773674135088461826, 8.132186467213601644042649985411, 8.463796393773269243471188882559, 9.526786095427157200586348637440, 10.87232677387080211304814565748, 12.86829873874676937079229131736

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