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Properties

Label 2-147-21.20-c3-0-3
Degree $2$
Conductor $147$
Sign $0.138 - 0.990i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.86i·2-s + (−2.76 + 4.40i)3-s − 0.235·4-s − 5.57·5-s + (12.6 + 7.92i)6-s − 22.2i·8-s + (−11.7 − 24.3i)9-s + 15.9i·10-s + 42.7i·11-s + (0.649 − 1.03i)12-s + 69.8i·13-s + (15.3 − 24.5i)15-s − 65.8·16-s − 67.5·17-s + (−69.7 + 33.6i)18-s + 79.3i·19-s + ⋯
L(s)  = 1  − 1.01i·2-s + (−0.531 + 0.846i)3-s − 0.0293·4-s − 0.498·5-s + (0.859 + 0.539i)6-s − 0.984i·8-s + (−0.434 − 0.900i)9-s + 0.505i·10-s + 1.17i·11-s + (0.0156 − 0.0248i)12-s + 1.48i·13-s + (0.264 − 0.421i)15-s − 1.02·16-s − 0.963·17-s + (−0.913 + 0.440i)18-s + 0.958i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.138 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.590796 + 0.513829i\)
\(L(\frac12)\) \(\approx\) \(0.590796 + 0.513829i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.76 - 4.40i)T \)
7 \( 1 \)
good2 \( 1 + 2.86iT - 8T^{2} \)
5 \( 1 + 5.57T + 125T^{2} \)
11 \( 1 - 42.7iT - 1.33e3T^{2} \)
13 \( 1 - 69.8iT - 2.19e3T^{2} \)
17 \( 1 + 67.5T + 4.91e3T^{2} \)
19 \( 1 - 79.3iT - 6.85e3T^{2} \)
23 \( 1 - 208. iT - 1.21e4T^{2} \)
29 \( 1 + 5.72iT - 2.43e4T^{2} \)
31 \( 1 + 193. iT - 2.97e4T^{2} \)
37 \( 1 - 163.T + 5.06e4T^{2} \)
41 \( 1 + 58.1T + 6.89e4T^{2} \)
43 \( 1 - 58.9T + 7.95e4T^{2} \)
47 \( 1 + 148.T + 1.03e5T^{2} \)
53 \( 1 - 100. iT - 1.48e5T^{2} \)
59 \( 1 + 738.T + 2.05e5T^{2} \)
61 \( 1 - 356. iT - 2.26e5T^{2} \)
67 \( 1 - 721.T + 3.00e5T^{2} \)
71 \( 1 + 308. iT - 3.57e5T^{2} \)
73 \( 1 + 827. iT - 3.89e5T^{2} \)
79 \( 1 - 467.T + 4.93e5T^{2} \)
83 \( 1 - 274.T + 5.71e5T^{2} \)
89 \( 1 + 541.T + 7.04e5T^{2} \)
97 \( 1 - 41.4iT - 9.12e5T^{2} \)
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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

−12.27944623564610137859961940643, −11.65548058602458630078862081768, −11.04785756100108170459466686374, −9.804827107654982205229933656687, −9.342142129303189495661622647624, −7.45361465872508974067157635659, −6.24058379083600070386559908082, −4.48830992515262956053618632486, −3.73363497441434304371039928885, −1.88083923313102091485676086034, 0.38793917049954226223174378913, 2.70524828251052004824558219845, 4.97218746500877538950427790749, 6.07220130057295433152083140785, 6.86305154989930638573449370234, 8.014636512528265839697242580663, 8.552137109009901569508418067452, 10.75115466145677001689826192125, 11.26646159033208090388083108115, 12.46298494565159743965547458871

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