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Properties

Label 2-147-7.6-c6-0-4
Degree $2$
Conductor $147$
Sign $-0.654 - 0.755i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.30·2-s + 15.5i·3-s + 22.6·4-s − 174. i·5-s + 145. i·6-s − 384.·8-s − 243·9-s − 1.62e3i·10-s − 185.·11-s + 353. i·12-s + 3.98e3i·13-s + 2.72e3·15-s − 5.03e3·16-s + 7.04e3i·17-s − 2.26e3·18-s + 4.69e3i·19-s + ⋯
L(s)  = 1  + 1.16·2-s + 0.577i·3-s + 0.353·4-s − 1.39i·5-s + 0.671i·6-s − 0.751·8-s − 0.333·9-s − 1.62i·10-s − 0.139·11-s + 0.204i·12-s + 1.81i·13-s + 0.807·15-s − 1.22·16-s + 1.43i·17-s − 0.387·18-s + 0.684i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ -0.654 - 0.755i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.476530579\)
\(L(\frac12)\) \(\approx\) \(1.476530579\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5iT \)
7 \( 1 \)
good2 \( 1 - 9.30T + 64T^{2} \)
5 \( 1 + 174. iT - 1.56e4T^{2} \)
11 \( 1 + 185.T + 1.77e6T^{2} \)
13 \( 1 - 3.98e3iT - 4.82e6T^{2} \)
17 \( 1 - 7.04e3iT - 2.41e7T^{2} \)
19 \( 1 - 4.69e3iT - 4.70e7T^{2} \)
23 \( 1 + 6.64e3T + 1.48e8T^{2} \)
29 \( 1 - 1.93e4T + 5.94e8T^{2} \)
31 \( 1 - 2.39e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.95e4T + 2.56e9T^{2} \)
41 \( 1 + 4.62e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.31e4T + 6.32e9T^{2} \)
47 \( 1 + 1.49e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.01e5T + 2.21e10T^{2} \)
59 \( 1 - 1.52e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.68e5iT - 5.15e10T^{2} \)
67 \( 1 - 6.11e4T + 9.04e10T^{2} \)
71 \( 1 + 4.85e5T + 1.28e11T^{2} \)
73 \( 1 + 8.92e3iT - 1.51e11T^{2} \)
79 \( 1 - 4.43e5T + 2.43e11T^{2} \)
83 \( 1 + 5.59e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.06e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.23e6iT - 8.32e11T^{2} \)
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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

−12.28389761866585344734358927712, −11.84277845851391709715655304156, −10.27533873659851342371052293805, −9.040914257115077907298836765447, −8.467502999329729820168870146313, −6.47325660420636328228935114149, −5.35489969038241749955078163824, −4.46104801411165626716960703991, −3.74078594854239528490807629975, −1.73665056638617538339498545074, 0.28340171946312187923465840759, 2.68527419596476466105958216331, 3.21148236792840809570734551321, 4.94712306691673411558724812452, 6.04628309845329196713605506331, 6.98157740713314722120778829110, 8.061592151909406841034607329690, 9.695969170521865408916840675245, 10.87187649110703147644462031991, 11.75509810509238273297199418898

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