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Properties

Label 2-147-21.20-c3-0-15
Degree $2$
Conductor $147$
Sign $0.355 + 0.934i$
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  − 1.05i·2-s + (−4.50 + 2.58i)3-s + 6.89·4-s − 10.2·5-s + (2.71 + 4.73i)6-s − 15.6i·8-s + (13.6 − 23.2i)9-s + 10.7i·10-s + 24.0i·11-s + (−31.0 + 17.8i)12-s − 49.4i·13-s + (46.1 − 26.4i)15-s + 38.7·16-s + 125.·17-s + (−24.4 − 14.3i)18-s − 95.5i·19-s + ⋯
L(s)  = 1  − 0.371i·2-s + (−0.867 + 0.497i)3-s + 0.862·4-s − 0.915·5-s + (0.184 + 0.322i)6-s − 0.691i·8-s + (0.505 − 0.862i)9-s + 0.339i·10-s + 0.659i·11-s + (−0.748 + 0.428i)12-s − 1.05i·13-s + (0.793 − 0.454i)15-s + 0.605·16-s + 1.79·17-s + (−0.320 − 0.187i)18-s − 1.15i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.355 + 0.934i$
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: \((2,\ 147,\ (\ :3/2),\ 0.355 + 0.934i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00123 - 0.690648i\)
\(L(\frac12)\) \(\approx\) \(1.00123 - 0.690648i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.50 - 2.58i)T \)
7 \( 1 \)
good2 \( 1 + 1.05iT - 8T^{2} \)
5 \( 1 + 10.2T + 125T^{2} \)
11 \( 1 - 24.0iT - 1.33e3T^{2} \)
13 \( 1 + 49.4iT - 2.19e3T^{2} \)
17 \( 1 - 125.T + 4.91e3T^{2} \)
19 \( 1 + 95.5iT - 6.85e3T^{2} \)
23 \( 1 + 185. iT - 1.21e4T^{2} \)
29 \( 1 + 40.6iT - 2.43e4T^{2} \)
31 \( 1 + 37.5iT - 2.97e4T^{2} \)
37 \( 1 - 283.T + 5.06e4T^{2} \)
41 \( 1 - 166.T + 6.89e4T^{2} \)
43 \( 1 + 411.T + 7.95e4T^{2} \)
47 \( 1 - 15.1T + 1.03e5T^{2} \)
53 \( 1 - 190. iT - 1.48e5T^{2} \)
59 \( 1 + 94.7T + 2.05e5T^{2} \)
61 \( 1 - 412. iT - 2.26e5T^{2} \)
67 \( 1 + 88.7T + 3.00e5T^{2} \)
71 \( 1 - 1.06e3iT - 3.57e5T^{2} \)
73 \( 1 + 828. iT - 3.89e5T^{2} \)
79 \( 1 - 256.T + 4.93e5T^{2} \)
83 \( 1 - 97.2T + 5.71e5T^{2} \)
89 \( 1 + 899.T + 7.04e5T^{2} \)
97 \( 1 + 398. iT - 9.12e5T^{2} \)
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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

−12.17656504860222432913685156961, −11.42451270477084880952807012275, −10.52201404513605120326921344684, −9.783582887086818337547258822470, −7.976562625989558842858658610232, −7.00423236355555896727668507733, −5.76278390482051042975422014515, −4.36831209834017574857851961007, −3.02109627553527277820869145905, −0.71681492003961513579747461593, 1.46170373636162417530462186607, 3.57500767372182680392513936494, 5.39394939358049808742536514684, 6.30741583420785281262577413429, 7.51189926311275081275928550485, 8.003107389518247278938634393447, 9.920323957326796313172495829855, 11.18814622144616934796427271665, 11.71820999150942060902853092703, 12.37318807096129575047603043608

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