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Properties

Label 2-147-147.134-c0-0-0
Degree $2$
Conductor $147$
Sign $0.801 - 0.598i$
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  + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)12-s + (−0.277 − 1.21i)13-s + (0.623 − 0.781i)16-s − 0.445·19-s + (0.623 − 0.781i)21-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)27-s + (0.623 + 0.781i)28-s − 1.80·31-s + (−0.222 − 0.974i)36-s + (1.62 + 0.781i)37-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)9-s + (−0.900 − 0.433i)12-s + (−0.277 − 1.21i)13-s + (0.623 − 0.781i)16-s − 0.445·19-s + (0.623 − 0.781i)21-s + (−0.222 + 0.974i)25-s + (−0.900 + 0.433i)27-s + (0.623 + 0.781i)28-s − 1.80·31-s + (−0.222 − 0.974i)36-s + (1.62 + 0.781i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(0.0733625\)
Root analytic conductor: \(0.270855\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :0),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6282494600\)
\(L(\frac12)\) \(\approx\) \(0.6282494600\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−13.29857992924145003477403868077, −12.88644052026460720973490967709, −11.14616323085050135116168379626, −10.13298575402331494128947906773, −9.399419790678040583871006831579, −8.245582394742496889386483699207, −7.43547382881228174632113469317, −5.37823019688536841080327108404, −4.19462357919782614420479415634, −3.21284140013096734124203552472, 2.17226243928963024610178804391, 4.02941450391131317545777145254, 5.62358814613490033814542175522, 6.74045206617909125982328451203, 8.202845077853546682508785480288, 9.056450249933553734618448309680, 9.716622130236253542420350773433, 11.39023318691897282337833823622, 12.51042067151936381790558136735, 13.12847629958155816232984445796

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