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Properties

Label 2-147-7.4-c5-0-5
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  + (−3.19 − 5.53i)2-s + (−4.5 + 7.79i)3-s + (−4.41 + 7.64i)4-s + (19.3 + 33.5i)5-s + 57.5·6-s − 148.·8-s + (−40.5 − 70.1i)9-s + (123. − 214. i)10-s + (288. − 499. i)11-s + (−39.7 − 68.8i)12-s − 391.·13-s − 348.·15-s + (614. + 1.06e3i)16-s + (−664. + 1.15e3i)17-s + (−258. + 448. i)18-s + (471. + 816. i)19-s + ⋯
L(s)  = 1  + (−0.564 − 0.978i)2-s + (−0.288 + 0.499i)3-s + (−0.137 + 0.238i)4-s + (0.346 + 0.599i)5-s + 0.652·6-s − 0.817·8-s + (−0.166 − 0.288i)9-s + (0.391 − 0.677i)10-s + (0.718 − 1.24i)11-s + (−0.0796 − 0.137i)12-s − 0.642·13-s − 0.399·15-s + (0.599 + 1.03i)16-s + (−0.557 + 0.966i)17-s + (−0.188 + 0.326i)18-s + (0.299 + 0.518i)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: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.123123242\)
\(L(\frac12)\) \(\approx\) \(1.123123242\)
\(L(\frac{7}{2})\) 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 \( 1 + (3.19 + 5.53i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-19.3 - 33.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-288. + 499. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 391.T + 3.71e5T^{2} \)
17 \( 1 + (664. - 1.15e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-471. - 816. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-816. - 1.41e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 1.46e3T + 2.05e7T^{2} \)
31 \( 1 + (1.95e3 - 3.38e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-8.15e3 - 1.41e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.31e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 + (3.40e3 + 5.90e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.00e3 + 1.74e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.57e4 + 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.05e4 - 3.55e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.52e4 - 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.99e4T + 1.80e9T^{2} \)
73 \( 1 + (2.78e4 - 4.82e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.15e4 - 5.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 4.55e4T + 3.93e9T^{2} \)
89 \( 1 + (-7.84e3 - 1.35e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 3.12e3T + 8.58e9T^{2} \)
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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

−11.69250782665497942619650955132, −11.06081738248087889613221781290, −10.25289123031483735137083630964, −9.415526967936590238651577273235, −8.411963297128130377393152855156, −6.58351155149557264993735963525, −5.68020654222288376945633244489, −3.81036429935956378440335847702, −2.63416789409053981479992505084, −1.06450552720894624047844550173, 0.57465756086427932340667051695, 2.35345011498053300397345305235, 4.61884884009651965233427604431, 5.83627597816343607064250837927, 7.05581357304159740614446503731, 7.53952866227194424061095752831, 9.104769651506203068980931387501, 9.458129814986588082066434409044, 11.23043404828620395191490979324, 12.30145314490107921091308028946

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