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Properties

Label 2-147-7.2-c5-0-10
Degree $2$
Conductor $147$
Sign $0.991 + 0.126i$
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  + (4.69 − 8.13i)2-s + (−4.5 − 7.79i)3-s + (−28.0 − 48.6i)4-s + (−35.8 + 62.1i)5-s − 84.5·6-s − 226.·8-s + (−40.5 + 70.1i)9-s + (336. + 583. i)10-s + (280. + 485. i)11-s + (−252. + 437. i)12-s − 533.·13-s + 645.·15-s + (−166. + 288. i)16-s + (502. + 870. i)17-s + (380. + 658. i)18-s + (684. − 1.18e3i)19-s + ⋯
L(s)  = 1  + (0.829 − 1.43i)2-s + (−0.288 − 0.499i)3-s + (−0.877 − 1.52i)4-s + (−0.641 + 1.11i)5-s − 0.958·6-s − 1.25·8-s + (−0.166 + 0.288i)9-s + (1.06 + 1.84i)10-s + (0.698 + 1.20i)11-s + (−0.506 + 0.877i)12-s − 0.875·13-s + 0.740·15-s + (−0.162 + 0.282i)16-s + (0.422 + 0.730i)17-s + (0.276 + 0.479i)18-s + (0.434 − 0.753i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.619999331\)
\(L(\frac12)\) \(\approx\) \(1.619999331\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.69 + 8.13i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (35.8 - 62.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-280. - 485. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 533.T + 3.71e5T^{2} \)
17 \( 1 + (-502. - 870. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-684. + 1.18e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.61e3 - 2.79e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 753.T + 2.05e7T^{2} \)
31 \( 1 + (-4.10e3 - 7.10e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.40e3 + 2.43e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 245.T + 1.15e8T^{2} \)
43 \( 1 + 1.75e4T + 1.47e8T^{2} \)
47 \( 1 + (8.17e3 - 1.41e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.48e4 - 2.56e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (5.17e3 + 8.96e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-477. + 826. i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-9.90e3 - 1.71e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.21e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.35e4 - 2.34e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.23e4 - 3.87e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.56e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.81e3 + 1.18e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.29e4T + 8.58e9T^{2} \)
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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

−12.00884119019060135401276668599, −11.45654325351979223473409307323, −10.43850107268601620212683452795, −9.618444116989085715666313718934, −7.63128611379144953824655800406, −6.71067307135083434897238490375, −5.09342162660238438750609300709, −3.87018363236734616704635388603, −2.73618795758857292681513818733, −1.48116380786680013415431719406, 0.45387600516148004167832666732, 3.62432673740447448215357559474, 4.61171376266579658724462522695, 5.45644156454027251844159661587, 6.53119287453275139723485155541, 7.917992529527696500683220263382, 8.584627435761080225252813523216, 9.863486911497925839211443504513, 11.63589166587677423826593192680, 12.25714634766738387334767937636

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