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Properties

Label 2-147-21.20-c5-0-1
Degree $2$
Conductor $147$
Sign $0.956 - 0.290i$
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  + 9.64i·2-s + (2.14 + 15.4i)3-s − 60.9·4-s − 95.0·5-s + (−148. + 20.6i)6-s − 279. i·8-s + (−233. + 66.2i)9-s − 916. i·10-s − 151. i·11-s + (−130. − 941. i)12-s + 901. i·13-s + (−203. − 1.46e3i)15-s + 741.·16-s + 488.·17-s + (−638. − 2.25e3i)18-s + 2.18e3i·19-s + ⋯
L(s)  = 1  + 1.70i·2-s + (0.137 + 0.990i)3-s − 1.90·4-s − 1.69·5-s + (−1.68 + 0.234i)6-s − 1.54i·8-s + (−0.962 + 0.272i)9-s − 2.89i·10-s − 0.378i·11-s + (−0.262 − 1.88i)12-s + 1.47i·13-s + (−0.233 − 1.68i)15-s + 0.723·16-s + 0.410·17-s + (−0.464 − 1.63i)18-s + 1.38i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.956 - 0.290i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1949943641\)
\(L(\frac12)\) \(\approx\) \(0.1949943641\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.14 - 15.4i)T \)
7 \( 1 \)
good2 \( 1 - 9.64iT - 32T^{2} \)
5 \( 1 + 95.0T + 3.12e3T^{2} \)
11 \( 1 + 151. iT - 1.61e5T^{2} \)
13 \( 1 - 901. iT - 3.71e5T^{2} \)
17 \( 1 - 488.T + 1.41e6T^{2} \)
19 \( 1 - 2.18e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.38e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.15e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.25e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.68e3T + 6.93e7T^{2} \)
41 \( 1 + 1.44e4T + 1.15e8T^{2} \)
43 \( 1 - 1.00e4T + 1.47e8T^{2} \)
47 \( 1 - 9.27e3T + 2.29e8T^{2} \)
53 \( 1 + 1.67e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.36e3T + 7.14e8T^{2} \)
61 \( 1 - 7.94e3iT - 8.44e8T^{2} \)
67 \( 1 - 3.39e4T + 1.35e9T^{2} \)
71 \( 1 - 4.31e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.15e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.01e4T + 3.07e9T^{2} \)
83 \( 1 - 2.96e3T + 3.93e9T^{2} \)
89 \( 1 + 8.44e4T + 5.58e9T^{2} \)
97 \( 1 + 6.07e4iT - 8.58e9T^{2} \)
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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

−13.83518360456644256917790742934, −12.06475550540941601039869401323, −11.25247292674789070149993227138, −9.750969991692466763558306541526, −8.603587657601926988944346199655, −8.025484448353332602653793781474, −6.96456387826468479911847481596, −5.62641665783228421827404831090, −4.36972398495463291028624877477, −3.71481598309543086417929809264, 0.087384460920671090615613557599, 0.950335565293602269814882859882, 2.70404829963886827422870644250, 3.55162327050222528500262617811, 4.96086479754365807017261542221, 7.08908462110778799684523394286, 8.079174036205984196984741769315, 8.975958545138577514749376044246, 10.56440810064627593333913947315, 11.25272186900630241918833598561

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