Properties

Label 2-147-7.6-c6-0-8
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  − 11.6·2-s + 15.5i·3-s + 70.9·4-s − 190. i·5-s − 181. i·6-s − 80.4·8-s − 243·9-s + 2.21e3i·10-s − 2.05e3·11-s + 1.10e3i·12-s + 3.05e3i·13-s + 2.97e3·15-s − 3.60e3·16-s − 2.85e3i·17-s + 2.82e3·18-s − 3.94e3i·19-s + ⋯
L(s)  = 1  − 1.45·2-s + 0.577i·3-s + 1.10·4-s − 1.52i·5-s − 0.838i·6-s − 0.157·8-s − 0.333·9-s + 2.21i·10-s − 1.54·11-s + 0.639i·12-s + 1.39i·13-s + 0.881·15-s − 0.879·16-s − 0.580i·17-s + 0.483·18-s − 0.575i·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

\(L(\frac{7}{2})\) \(\approx\) \(0.4808458485\)
\(L(\frac12)\) \(\approx\) \(0.4808458485\)
\(L(4)\) 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 \( 1 + 11.6T + 64T^{2} \)
5 \( 1 + 190. iT - 1.56e4T^{2} \)
11 \( 1 + 2.05e3T + 1.77e6T^{2} \)
13 \( 1 - 3.05e3iT - 4.82e6T^{2} \)
17 \( 1 + 2.85e3iT - 2.41e7T^{2} \)
19 \( 1 + 3.94e3iT - 4.70e7T^{2} \)
23 \( 1 - 660.T + 1.48e8T^{2} \)
29 \( 1 + 9.28e3T + 5.94e8T^{2} \)
31 \( 1 + 2.80e3iT - 8.87e8T^{2} \)
37 \( 1 + 3.69e4T + 2.56e9T^{2} \)
41 \( 1 + 6.79e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.23e4T + 6.32e9T^{2} \)
47 \( 1 - 1.47e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.19e5T + 2.21e10T^{2} \)
59 \( 1 + 1.92e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.32e5iT - 5.15e10T^{2} \)
67 \( 1 - 3.49e5T + 9.04e10T^{2} \)
71 \( 1 - 3.05e5T + 1.28e11T^{2} \)
73 \( 1 - 2.36e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.86e5T + 2.43e11T^{2} \)
83 \( 1 - 1.06e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.30e4iT - 4.96e11T^{2} \)
97 \( 1 - 2.05e5iT - 8.32e11T^{2} \)
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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

−11.74952367677392697488588098798, −10.75827301695586625939022674734, −9.684875960524517336173884969316, −9.024847445711395712707358853816, −8.314090936667349930962434655282, −7.19916782457905686897216874884, −5.32494517350602127565005684863, −4.41274518773843202489639465003, −2.19276037064304946665000035523, −0.73408990079405957977637201776, 0.35608495680581234003829326691, 2.08356913050186560405149184027, 3.13989877508308713327576153302, 5.62299572727144211423161531249, 6.91402208462778974015499238130, 7.73234855061136440431400401269, 8.363452762014134459974183097385, 10.08985798266490371048751319394, 10.45402322318675725541209075501, 11.27462504042294674074909083388

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