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Properties

Label 2-147-21.20-c3-0-17
Degree $2$
Conductor $147$
Sign $0.936 - 0.350i$
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  − 0.222i·2-s + (−3.69 + 3.65i)3-s + 7.95·4-s + 18.7·5-s + (0.814 + 0.823i)6-s − 3.55i·8-s + (0.292 − 26.9i)9-s − 4.16i·10-s − 45.8i·11-s + (−29.3 + 29.0i)12-s + 59.3i·13-s + (−69.0 + 68.3i)15-s + 62.8·16-s + 28.9·17-s + (−6.01 − 0.0652i)18-s − 38.9i·19-s + ⋯
L(s)  = 1  − 0.0788i·2-s + (−0.710 + 0.703i)3-s + 0.993·4-s + 1.67·5-s + (0.0554 + 0.0560i)6-s − 0.157i·8-s + (0.0108 − 0.999i)9-s − 0.131i·10-s − 1.25i·11-s + (−0.706 + 0.698i)12-s + 1.26i·13-s + (−1.18 + 1.17i)15-s + 0.981·16-s + 0.412·17-s + (−0.0788 − 0.000854i)18-s − 0.470i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.936 - 0.350i$
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: \((2,\ 147,\ (\ :3/2),\ 0.936 - 0.350i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.14144 + 0.388024i\)
\(L(\frac12)\) \(\approx\) \(2.14144 + 0.388024i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.69 - 3.65i)T \)
7 \( 1 \)
good2 \( 1 + 0.222iT - 8T^{2} \)
5 \( 1 - 18.7T + 125T^{2} \)
11 \( 1 + 45.8iT - 1.33e3T^{2} \)
13 \( 1 - 59.3iT - 2.19e3T^{2} \)
17 \( 1 - 28.9T + 4.91e3T^{2} \)
19 \( 1 + 38.9iT - 6.85e3T^{2} \)
23 \( 1 - 1.79iT - 1.21e4T^{2} \)
29 \( 1 - 148. iT - 2.43e4T^{2} \)
31 \( 1 - 104. iT - 2.97e4T^{2} \)
37 \( 1 + 24.1T + 5.06e4T^{2} \)
41 \( 1 + 254.T + 6.89e4T^{2} \)
43 \( 1 + 59.6T + 7.95e4T^{2} \)
47 \( 1 + 262.T + 1.03e5T^{2} \)
53 \( 1 - 381. iT - 1.48e5T^{2} \)
59 \( 1 + 371.T + 2.05e5T^{2} \)
61 \( 1 + 696. iT - 2.26e5T^{2} \)
67 \( 1 - 434.T + 3.00e5T^{2} \)
71 \( 1 + 824. iT - 3.57e5T^{2} \)
73 \( 1 + 0.395iT - 3.89e5T^{2} \)
79 \( 1 + 731.T + 4.93e5T^{2} \)
83 \( 1 + 586.T + 5.71e5T^{2} \)
89 \( 1 - 440.T + 7.04e5T^{2} \)
97 \( 1 - 634. iT - 9.12e5T^{2} \)
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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

−12.46546226408729042616760720932, −11.38008396842298815306847272962, −10.68630051662164030475049909676, −9.775155066085783275401616305742, −8.845370652800725702806139748784, −6.76353210333728753984951893459, −6.11927189388170199317777188824, −5.14184659271883621482941447838, −3.17349508415062647142599669295, −1.52505602785168986619774334332, 1.49800311171418479746270703873, 2.50435286707437247467456587351, 5.27368377323429033503200653052, 6.00715889726234648255543807692, 6.94231484901361835011306097175, 7.999319884012400199780677541988, 9.957118692270334199105299251384, 10.31552621387012939423363762517, 11.62059861824870612741355594872, 12.62725690195539351689399263316

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