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 $0$

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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