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Properties

Label 2-147-147.92-c0-0-0
Degree $2$
Conductor $147$
Sign $0.518 + 0.855i$
Analytic cond. $0.0733625$
Root an. cond. $0.270855$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)12-s + (−1.12 + 1.40i)13-s + (−0.900 + 0.433i)16-s + 1.24·19-s + (−0.900 + 0.433i)21-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (−0.900 − 0.433i)28-s − 0.445·31-s + (0.623 − 0.781i)36-s + (0.0990 − 0.433i)37-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)12-s + (−1.12 + 1.40i)13-s + (−0.900 + 0.433i)16-s + 1.24·19-s + (−0.900 + 0.433i)21-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (−0.900 − 0.433i)28-s − 0.445·31-s + (0.623 − 0.781i)36-s + (0.0990 − 0.433i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.518 + 0.855i$
Analytic conductor: \(0.0733625\)
Root analytic conductor: \(0.270855\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :0),\ 0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5030964000\)
\(L(\frac12)\) \(\approx\) \(0.5030964000\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−13.25732861719228450539883026316, −11.88178611580306245556353757463, −11.20449055841784289390214617734, −10.22638645859426854822911819067, −9.283901789721509241191267639385, −7.50644733429563399858112453541, −6.72254909617020985484224982718, −5.32065799767493430308145670719, −4.52402876912039506229677954468, −1.58715089653729890030981512683, 3.03019981309096679580850642751, 4.70153546443400454209523227185, 5.54992912851620471209409676054, 7.18338939087353266449379690246, 8.216512192743882980429725440452, 9.421846044331542266949467387515, 10.50552222409029036954061636298, 11.76800986394345354496296409303, 12.19107042111006026078096654767, 13.14212526949402757005998691217

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