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Properties

Label 2-147-21.20-c5-0-51
Degree $2$
Conductor $147$
Sign $-0.967 + 0.251i$
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  − 3.81i·2-s + (6.23 + 14.2i)3-s + 17.4·4-s − 63.5·5-s + (54.4 − 23.7i)6-s − 188. i·8-s + (−165. + 178. i)9-s + 242. i·10-s − 192. i·11-s + (108. + 249. i)12-s − 82.7i·13-s + (−396. − 908. i)15-s − 160.·16-s − 1.84e3·17-s + (679. + 630. i)18-s + 2.02e3i·19-s + ⋯
L(s)  = 1  − 0.674i·2-s + (0.399 + 0.916i)3-s + 0.545·4-s − 1.13·5-s + (0.617 − 0.269i)6-s − 1.04i·8-s + (−0.680 + 0.732i)9-s + 0.766i·10-s − 0.480i·11-s + (0.218 + 0.500i)12-s − 0.135i·13-s + (−0.454 − 1.04i)15-s − 0.156·16-s − 1.55·17-s + (0.494 + 0.458i)18-s + 1.28i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.967 + 0.251i$
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: \((2,\ 147,\ (\ :5/2),\ -0.967 + 0.251i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4523715248\)
\(L(\frac12)\) \(\approx\) \(0.4523715248\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.23 - 14.2i)T \)
7 \( 1 \)
good2 \( 1 + 3.81iT - 32T^{2} \)
5 \( 1 + 63.5T + 3.12e3T^{2} \)
11 \( 1 + 192. iT - 1.61e5T^{2} \)
13 \( 1 + 82.7iT - 3.71e5T^{2} \)
17 \( 1 + 1.84e3T + 1.41e6T^{2} \)
19 \( 1 - 2.02e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.75e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.54e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.70e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.57e4T + 6.93e7T^{2} \)
41 \( 1 - 1.05e4T + 1.15e8T^{2} \)
43 \( 1 + 6.69e3T + 1.47e8T^{2} \)
47 \( 1 + 1.69e4T + 2.29e8T^{2} \)
53 \( 1 - 2.94e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.26e4T + 7.14e8T^{2} \)
61 \( 1 - 7.69e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.99e4T + 1.35e9T^{2} \)
71 \( 1 + 1.68e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.41e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.57e4T + 3.07e9T^{2} \)
83 \( 1 - 2.44e4T + 3.93e9T^{2} \)
89 \( 1 - 8.01e4T + 5.58e9T^{2} \)
97 \( 1 - 3.67e3iT - 8.58e9T^{2} \)
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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

−11.42717851051883528162308656010, −10.84632891530274934656744790728, −9.871778378726393316459055735114, −8.594750028203286980210617145804, −7.67491963281053831760773510835, −6.17930128826800683578759301259, −4.35377280502563272471990915579, −3.57359990705059564414238561736, −2.29566954875219643892025599155, −0.13148125011760699108762306895, 1.84339159082037246937118006494, 3.27829690076727759015830024729, 5.02197694584335133298440363640, 6.78306245023037276524604292394, 7.06052345282809702523436997390, 8.180177870205306572573273370613, 8.998101028246851432778938790007, 10.98025227556983554331541374599, 11.64533702431995738613467817844, 12.56683653979411176798594939517

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