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Properties

Label 2-147-7.3-c6-0-23
Degree $2$
Conductor $147$
Sign $0.553 - 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.25 + 3.90i)2-s + (13.5 + 7.79i)3-s + (21.8 − 37.8i)4-s + (−53.9 + 31.1i)5-s + 70.2i·6-s + 485.·8-s + (121.5 + 210. i)9-s + (−243. − 140. i)10-s + (−9.71 + 16.8i)11-s + (589. − 340. i)12-s + 1.64e3i·13-s − 970.·15-s + (−304. − 527. i)16-s + (186. + 107. i)17-s + (−547. + 948. i)18-s + (8.23e3 − 4.75e3i)19-s + ⋯
L(s)  = 1  + (0.281 + 0.487i)2-s + (0.5 + 0.288i)3-s + (0.341 − 0.591i)4-s + (−0.431 + 0.249i)5-s + 0.325i·6-s + 0.947·8-s + (0.166 + 0.288i)9-s + (−0.243 − 0.140i)10-s + (−0.00730 + 0.0126i)11-s + (0.341 − 0.197i)12-s + 0.747i·13-s − 0.287·15-s + (−0.0743 − 0.128i)16-s + (0.0378 + 0.0218i)17-s + (−0.0938 + 0.162i)18-s + (1.20 − 0.693i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.129530677\)
\(L(\frac12)\) \(\approx\) \(3.129530677\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-2.25 - 3.90i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (53.9 - 31.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (9.71 - 16.8i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 1.64e3iT - 4.82e6T^{2} \)
17 \( 1 + (-186. - 107. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-8.23e3 + 4.75e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-7.22e3 - 1.25e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 4.30e4T + 5.94e8T^{2} \)
31 \( 1 + (-7.79e3 - 4.50e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.65e4 - 2.86e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 7.37e4iT - 4.75e9T^{2} \)
43 \( 1 - 4.76e3T + 6.32e9T^{2} \)
47 \( 1 + (-6.37e4 + 3.68e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.19e5 - 2.06e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-2.82e5 - 1.63e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (3.50e5 - 2.02e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.09e5 - 1.90e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 3.50e5T + 1.28e11T^{2} \)
73 \( 1 + (-1.74e5 - 1.00e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.97e5 + 3.42e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 1.32e4iT - 3.26e11T^{2} \)
89 \( 1 + (-1.99e5 + 1.15e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 6.62e5iT - 8.32e11T^{2} \)
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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

−11.93311190804802919532962795718, −11.07864807283480076725977439551, −10.00940016935195960395016663217, −9.022163876833879800309599086895, −7.59971595809041644967188149001, −6.86514093550617357934456344901, −5.47835569277458147118227624357, −4.35544499694806567107449314801, −2.88575210105881329088266846341, −1.26332220040725432809725293153, 0.948111364165332860178384524742, 2.53913571608059017078815475841, 3.51063412931355057159362743228, 4.76878430837055829669058143480, 6.53470454557621599239630292790, 7.80565596550734353595141221845, 8.312438569570241030662485483235, 9.831831399743144557937394822887, 10.94216934545344383805649022463, 12.06796650183988200881271641031

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