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Properties

Label 2-147-21.20-c7-0-32
Degree $2$
Conductor $147$
Sign $0.0765 - 0.997i$
Analytic cond. $45.9205$
Root an. cond. $6.77647$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.39i·2-s + (33.2 − 32.9i)3-s + 73.2·4-s − 203.·5-s + (243. + 245. i)6-s + 1.48e3i·8-s + (21.5 − 2.18e3i)9-s − 1.50e3i·10-s + 1.94e3i·11-s + (2.43e3 − 2.41e3i)12-s + 7.87e3i·13-s + (−6.74e3 + 6.68e3i)15-s − 1.63e3·16-s + 9.23e3·17-s + (1.61e4 + 159. i)18-s − 1.04e4i·19-s + ⋯
L(s)  = 1  + 0.653i·2-s + (0.710 − 0.703i)3-s + 0.572·4-s − 0.726·5-s + (0.460 + 0.464i)6-s + 1.02i·8-s + (0.00986 − 0.999i)9-s − 0.475i·10-s + 0.440i·11-s + (0.406 − 0.402i)12-s + 0.994i·13-s + (−0.516 + 0.511i)15-s − 0.0999·16-s + 0.455·17-s + (0.653 + 0.00645i)18-s − 0.348i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0765 - 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0765 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0765 - 0.997i$
Analytic conductor: \(45.9205\)
Root analytic conductor: \(6.77647\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :7/2),\ 0.0765 - 0.997i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.545151624\)
\(L(\frac12)\) \(\approx\) \(2.545151624\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.2 + 32.9i)T \)
7 \( 1 \)
good2 \( 1 - 7.39iT - 128T^{2} \)
5 \( 1 + 203.T + 7.81e4T^{2} \)
11 \( 1 - 1.94e3iT - 1.94e7T^{2} \)
13 \( 1 - 7.87e3iT - 6.27e7T^{2} \)
17 \( 1 - 9.23e3T + 4.10e8T^{2} \)
19 \( 1 + 1.04e4iT - 8.93e8T^{2} \)
23 \( 1 - 8.18e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.95e5iT - 1.72e10T^{2} \)
31 \( 1 + 9.74e4iT - 2.75e10T^{2} \)
37 \( 1 - 5.70e5T + 9.49e10T^{2} \)
41 \( 1 - 3.39e5T + 1.94e11T^{2} \)
43 \( 1 + 3.56e5T + 2.71e11T^{2} \)
47 \( 1 - 7.66e5T + 5.06e11T^{2} \)
53 \( 1 - 1.58e5iT - 1.17e12T^{2} \)
59 \( 1 - 2.56e6T + 2.48e12T^{2} \)
61 \( 1 - 1.72e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.20e6T + 6.06e12T^{2} \)
71 \( 1 - 5.29e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.54e6iT - 1.10e13T^{2} \)
79 \( 1 - 7.44e5T + 1.92e13T^{2} \)
83 \( 1 + 5.20e6T + 2.71e13T^{2} \)
89 \( 1 + 1.01e7T + 4.42e13T^{2} \)
97 \( 1 - 3.36e6iT - 8.07e13T^{2} \)
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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

−11.88836676757047177253694301571, −11.33076391560013467287545909928, −9.633538088798343892919699679872, −8.511481126714817238907042977291, −7.48817991577464303709559165516, −7.04093042163476745470073907045, −5.75855830446232668011802352925, −4.02243390275498725409319572998, −2.63107002116266507349211473725, −1.37513206023610385876454705141, 0.63143162940159067180192360187, 2.38514132542568323328885437655, 3.33105325017937273600926404274, 4.31276240977910226227946178596, 5.97651005570504357939957627825, 7.59283022456846665853159119743, 8.301092251136788732010791178481, 9.718806219017855577036669842125, 10.50596294407260288855238871694, 11.33158343475732746017645988811

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