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Properties

Label 2-147-21.20-c3-0-22
Degree $2$
Conductor $147$
Sign $0.945 - 0.326i$
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  + 2.58i·2-s + (2.60 − 4.49i)3-s + 1.30·4-s + 16.1·5-s + (11.6 + 6.73i)6-s + 24.0i·8-s + (−13.4 − 23.4i)9-s + 41.7i·10-s − 35.5i·11-s + (3.39 − 5.87i)12-s − 7.40i·13-s + (41.9 − 72.4i)15-s − 51.8·16-s + 28.9·17-s + (60.5 − 34.8i)18-s + 35.1i·19-s + ⋯
L(s)  = 1  + 0.914i·2-s + (0.500 − 0.865i)3-s + 0.163·4-s + 1.44·5-s + (0.791 + 0.458i)6-s + 1.06i·8-s + (−0.498 − 0.867i)9-s + 1.31i·10-s − 0.975i·11-s + (0.0817 − 0.141i)12-s − 0.158i·13-s + (0.722 − 1.24i)15-s − 0.810·16-s + 0.412·17-s + (0.793 − 0.455i)18-s + 0.424i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.945 - 0.326i$
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: \((2,\ 147,\ (\ :3/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.64372 + 0.443578i\)
\(L(\frac12)\) \(\approx\) \(2.64372 + 0.443578i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.60 + 4.49i)T \)
7 \( 1 \)
good2 \( 1 - 2.58iT - 8T^{2} \)
5 \( 1 - 16.1T + 125T^{2} \)
11 \( 1 + 35.5iT - 1.33e3T^{2} \)
13 \( 1 + 7.40iT - 2.19e3T^{2} \)
17 \( 1 - 28.9T + 4.91e3T^{2} \)
19 \( 1 - 35.1iT - 6.85e3T^{2} \)
23 \( 1 - 55.4iT - 1.21e4T^{2} \)
29 \( 1 + 68.1iT - 2.43e4T^{2} \)
31 \( 1 - 178. iT - 2.97e4T^{2} \)
37 \( 1 + 233.T + 5.06e4T^{2} \)
41 \( 1 - 370.T + 6.89e4T^{2} \)
43 \( 1 + 187.T + 7.95e4T^{2} \)
47 \( 1 - 174.T + 1.03e5T^{2} \)
53 \( 1 - 272. iT - 1.48e5T^{2} \)
59 \( 1 - 96.8T + 2.05e5T^{2} \)
61 \( 1 + 385. iT - 2.26e5T^{2} \)
67 \( 1 + 1.01e3T + 3.00e5T^{2} \)
71 \( 1 + 125. iT - 3.57e5T^{2} \)
73 \( 1 + 225. iT - 3.89e5T^{2} \)
79 \( 1 + 1.06e3T + 4.93e5T^{2} \)
83 \( 1 + 601.T + 5.71e5T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 - 327. iT - 9.12e5T^{2} \)
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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

−12.97789416791434908033728179076, −11.79855571523008641232817486381, −10.51865256364668570507047281425, −9.197367244292218906594881334008, −8.282663190684389873484206717699, −7.19671870935127085798065890733, −6.12431268211630971668610239460, −5.57956309014487267696872622332, −2.93509229163369546656775984105, −1.59333761152944019449775745938, 1.83411433169953708913262486501, 2.81069709100425223123977288489, 4.36511763763900482305210713198, 5.75676351430161776878071920748, 7.16947432253756454767930433972, 8.918029035532858419244590502888, 9.837251476749226921511216936168, 10.23760081333176737175936886535, 11.29377468331514643024833131513, 12.55603583807526259835488090259

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