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Properties

Label 2-147-7.4-c5-0-22
Degree $2$
Conductor $147$
Sign $-0.386 + 0.922i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (4.5 − 7.79i)3-s + (15.5 − 26.8i)4-s + (17 + 29.4i)5-s − 9·6-s − 63·8-s + (−40.5 − 70.1i)9-s + (16.9 − 29.4i)10-s + (170 − 294. i)11-s + (−139.5 − 241. i)12-s + 454·13-s + 306·15-s + (−464.5 − 804. i)16-s + (399 − 691. i)17-s + (−40.5 + 70.1i)18-s + (−446 − 772. i)19-s + ⋯
L(s)  = 1  + (−0.0883 − 0.153i)2-s + (0.288 − 0.499i)3-s + (0.484 − 0.838i)4-s + (0.304 + 0.526i)5-s − 0.102·6-s − 0.348·8-s + (−0.166 − 0.288i)9-s + (0.0537 − 0.0931i)10-s + (0.423 − 0.733i)11-s + (−0.279 − 0.484i)12-s + 0.745·13-s + 0.351·15-s + (−0.453 − 0.785i)16-s + (0.334 − 0.579i)17-s + (−0.0294 + 0.0510i)18-s + (−0.283 − 0.490i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.266184886\)
\(L(\frac12)\) \(\approx\) \(2.266184886\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-17 - 29.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-170 + 294. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 454T + 3.71e5T^{2} \)
17 \( 1 + (-399 + 691. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (446 + 772. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.59e3 - 2.76e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 8.24e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.24e3 + 2.16e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.89e3 + 8.48e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.98e4T + 1.15e8T^{2} \)
43 \( 1 + 1.72e4T + 1.47e8T^{2} \)
47 \( 1 + (4.46e3 + 7.73e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (75 - 129. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.11e4 + 3.67e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (7.37e3 + 1.27e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-838 + 1.45e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.45e4T + 1.80e9T^{2} \)
73 \( 1 + (3.91e4 - 6.78e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.13e3 - 1.96e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 3.77e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.86e4 - 1.01e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.00e4T + 8.58e9T^{2} \)
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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

−11.46185058634540938349271985442, −11.04683385692800014174231081683, −9.759684604346111563874226656577, −8.859782024463065321493213918077, −7.37102860987291776611890731681, −6.41132485697829752175078152583, −5.49693741508349489769407815924, −3.40301642306557464678778925580, −2.08247390718390313233196119436, −0.76667496425119150520312119182, 1.71804856053153784102608933785, 3.30980044719199808460636837565, 4.45789763170981101844147039679, 5.99840586250065689225929036539, 7.25496542246421484664806699073, 8.429342519160791736982855892701, 9.145710049258621568770070955734, 10.40239734879337093154202076735, 11.46494446111996955188613008636, 12.58533218067522064031375841121

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