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Properties

Label 2-147-21.20-c3-0-35
Degree $2$
Conductor $147$
Sign $0.841 - 0.539i$
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  − 5.37i·2-s + (−2.08 − 4.75i)3-s − 20.8·4-s + 2.78·5-s + (−25.5 + 11.2i)6-s + 69.3i·8-s + (−18.3 + 19.8i)9-s − 14.9i·10-s + 17.5i·11-s + (43.5 + 99.4i)12-s − 47.8i·13-s + (−5.81 − 13.2i)15-s + 205.·16-s − 89.4·17-s + (106. + 98.3i)18-s − 42.2i·19-s + ⋯
L(s)  = 1  − 1.90i·2-s + (−0.401 − 0.915i)3-s − 2.61·4-s + 0.249·5-s + (−1.74 + 0.762i)6-s + 3.06i·8-s + (−0.677 + 0.735i)9-s − 0.473i·10-s + 0.481i·11-s + (1.04 + 2.39i)12-s − 1.02i·13-s + (−0.100 − 0.228i)15-s + 3.21·16-s − 1.27·17-s + (1.39 + 1.28i)18-s − 0.510i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.841 - 0.539i$
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: \((2,\ 147,\ (\ :3/2),\ 0.841 - 0.539i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.280638 + 0.0822421i\)
\(L(\frac12)\) \(\approx\) \(0.280638 + 0.0822421i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.08 + 4.75i)T \)
7 \( 1 \)
good2 \( 1 + 5.37iT - 8T^{2} \)
5 \( 1 - 2.78T + 125T^{2} \)
11 \( 1 - 17.5iT - 1.33e3T^{2} \)
13 \( 1 + 47.8iT - 2.19e3T^{2} \)
17 \( 1 + 89.4T + 4.91e3T^{2} \)
19 \( 1 + 42.2iT - 6.85e3T^{2} \)
23 \( 1 - 87.6iT - 1.21e4T^{2} \)
29 \( 1 + 40.8iT - 2.43e4T^{2} \)
31 \( 1 + 95.6iT - 2.97e4T^{2} \)
37 \( 1 + 64.5T + 5.06e4T^{2} \)
41 \( 1 - 403.T + 6.89e4T^{2} \)
43 \( 1 + 230.T + 7.95e4T^{2} \)
47 \( 1 + 365.T + 1.03e5T^{2} \)
53 \( 1 - 598. iT - 1.48e5T^{2} \)
59 \( 1 + 236.T + 2.05e5T^{2} \)
61 \( 1 - 430. iT - 2.26e5T^{2} \)
67 \( 1 + 428.T + 3.00e5T^{2} \)
71 \( 1 + 519. iT - 3.57e5T^{2} \)
73 \( 1 + 764. iT - 3.89e5T^{2} \)
79 \( 1 + 227.T + 4.93e5T^{2} \)
83 \( 1 + 1.13e3T + 5.71e5T^{2} \)
89 \( 1 + 1.13e3T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3iT - 9.12e5T^{2} \)
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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

−11.62075359998600227444631017672, −10.95871611261229522338655549458, −9.938578884932182106890663232748, −8.855328310629823930161313334570, −7.63148021232665564677512363088, −5.82007362294415497878478291494, −4.52836093399042612744680753427, −2.82059221615190546445660519975, −1.68469792340236696267768938165, −0.15110160268305911527600071334, 3.97049069948735871500573141960, 4.92710910199368791345363719965, 6.06362768241413878234655862410, 6.79482855669934488055806901825, 8.345852615919036906499218746160, 9.097865800975956824149706916075, 10.01620245073050989584600972854, 11.38583218848288886915512233968, 12.90127031284060428619919588775, 14.04657631613488927995091926184

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