Properties

Label 2-147-21.20-c3-0-5
Degree $2$
Conductor $147$
Sign $-0.112 + 0.993i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.54i·2-s + (2.93 + 4.28i)3-s − 12.6·4-s − 11.6·5-s + (−19.4 + 13.3i)6-s − 21.1i·8-s + (−9.72 + 25.1i)9-s − 52.7i·10-s + 17.9i·11-s + (−37.2 − 54.2i)12-s − 62.4i·13-s + (−34.1 − 49.7i)15-s − 4.97·16-s + 21.4·17-s + (−114. − 44.1i)18-s + 10.9i·19-s + ⋯
L(s)  = 1  + 1.60i·2-s + (0.565 + 0.824i)3-s − 1.58·4-s − 1.03·5-s + (−1.32 + 0.909i)6-s − 0.936i·8-s + (−0.360 + 0.932i)9-s − 1.66i·10-s + 0.491i·11-s + (−0.895 − 1.30i)12-s − 1.33i·13-s + (−0.587 − 0.855i)15-s − 0.0777·16-s + 0.305·17-s + (−1.49 − 0.578i)18-s + 0.132i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.112 + 0.993i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.112 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.624761 - 0.699294i\)
\(L(\frac12)\) \(\approx\) \(0.624761 - 0.699294i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.93 - 4.28i)T \)
7 \( 1 \)
good2 \( 1 - 4.54iT - 8T^{2} \)
5 \( 1 + 11.6T + 125T^{2} \)
11 \( 1 - 17.9iT - 1.33e3T^{2} \)
13 \( 1 + 62.4iT - 2.19e3T^{2} \)
17 \( 1 - 21.4T + 4.91e3T^{2} \)
19 \( 1 - 10.9iT - 6.85e3T^{2} \)
23 \( 1 - 69.0iT - 1.21e4T^{2} \)
29 \( 1 - 265. iT - 2.43e4T^{2} \)
31 \( 1 + 10.2iT - 2.97e4T^{2} \)
37 \( 1 - 41.6T + 5.06e4T^{2} \)
41 \( 1 - 31.0T + 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 + 163.T + 1.03e5T^{2} \)
53 \( 1 - 527. iT - 1.48e5T^{2} \)
59 \( 1 + 411.T + 2.05e5T^{2} \)
61 \( 1 - 258. iT - 2.26e5T^{2} \)
67 \( 1 - 323.T + 3.00e5T^{2} \)
71 \( 1 + 45.4iT - 3.57e5T^{2} \)
73 \( 1 - 562. iT - 3.89e5T^{2} \)
79 \( 1 - 289.T + 4.93e5T^{2} \)
83 \( 1 + 448.T + 5.71e5T^{2} \)
89 \( 1 + 561.T + 7.04e5T^{2} \)
97 \( 1 - 214. iT - 9.12e5T^{2} \)
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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

−13.78164357806303729583224017605, −12.59844703167511382982006276865, −11.14819884462100450862129652029, −9.956539385834118507054819284762, −8.770600821417188795038406821239, −7.941258231586094200519233450647, −7.28598424772383993308120425194, −5.62375830095227687252299993367, −4.63865493663607330155469255830, −3.36559498331477967299174032619, 0.42576483022464004482419783046, 1.99204651913730713804173766143, 3.33137254216542675239001762537, 4.31847883148021139753607808685, 6.56095679665319548315243427630, 7.898705563252926836667860081758, 8.849922706590220429555954702969, 9.868050640338275001804760565111, 11.38132208091442766782411213737, 11.68807995215149371682035264428

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