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Properties

Label 2-147-7.6-c6-0-14
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  + 5.52·2-s + 15.5i·3-s − 33.4·4-s − 66.9i·5-s + 86.1i·6-s − 538.·8-s − 243·9-s − 370. i·10-s + 1.72e3·11-s − 521. i·12-s − 2.80e3i·13-s + 1.04e3·15-s − 836.·16-s + 6.14e3i·17-s − 1.34e3·18-s + 8.93e3i·19-s + ⋯
L(s)  = 1  + 0.690·2-s + 0.577i·3-s − 0.522·4-s − 0.535i·5-s + 0.398i·6-s − 1.05·8-s − 0.333·9-s − 0.370i·10-s + 1.29·11-s − 0.301i·12-s − 1.27i·13-s + 0.309·15-s − 0.204·16-s + 1.25i·17-s − 0.230·18-s + 1.30i·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

\(L(\frac{7}{2})\) \(\approx\) \(2.279421643\)
\(L(\frac12)\) \(\approx\) \(2.279421643\)
\(L(4)\) 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 \( 1 - 5.52T + 64T^{2} \)
5 \( 1 + 66.9iT - 1.56e4T^{2} \)
11 \( 1 - 1.72e3T + 1.77e6T^{2} \)
13 \( 1 + 2.80e3iT - 4.82e6T^{2} \)
17 \( 1 - 6.14e3iT - 2.41e7T^{2} \)
19 \( 1 - 8.93e3iT - 4.70e7T^{2} \)
23 \( 1 - 9.90e3T + 1.48e8T^{2} \)
29 \( 1 + 1.36e4T + 5.94e8T^{2} \)
31 \( 1 - 2.52e4iT - 8.87e8T^{2} \)
37 \( 1 - 2.27e4T + 2.56e9T^{2} \)
41 \( 1 - 3.78e4iT - 4.75e9T^{2} \)
43 \( 1 - 7.36e4T + 6.32e9T^{2} \)
47 \( 1 - 1.39e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.65e4T + 2.21e10T^{2} \)
59 \( 1 - 7.15e4iT - 4.21e10T^{2} \)
61 \( 1 + 2.63e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.48e5T + 9.04e10T^{2} \)
71 \( 1 - 4.65e5T + 1.28e11T^{2} \)
73 \( 1 - 3.33e5iT - 1.51e11T^{2} \)
79 \( 1 + 4.10e5T + 2.43e11T^{2} \)
83 \( 1 + 7.19e4iT - 3.26e11T^{2} \)
89 \( 1 + 1.63e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.51e5iT - 8.32e11T^{2} \)
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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

−12.54336592190719072952791355766, −11.11979289514418207047127801261, −9.945869378320283691978486186347, −8.987410158171263680327308078748, −8.144255344538956302426679961865, −6.23358266558794925297047354359, −5.30631344488908243697894922401, −4.18175963159195489537073850209, −3.31408060215772836685577922180, −1.06402532321720691182323759061, 0.71048645928090666030012604580, 2.53135947836533200955198029359, 3.88633271593538513934564387767, 5.04032353947480114089377924706, 6.48695713739567690187709946364, 7.13691391060641285250202103218, 8.932163967214271594128709777311, 9.395788631681376380908224976225, 11.26883467894353173234445956218, 11.81030684484138409299187211339

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