Properties

Label 2-147-7.3-c6-0-37
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  + (5.80 + 10.0i)2-s + (13.5 + 7.79i)3-s + (−35.4 + 61.4i)4-s + (165. − 95.4i)5-s + 181. i·6-s − 80.4·8-s + (121.5 + 210. i)9-s + (1.92e3 + 1.10e3i)10-s + (1.02e3 − 1.77e3i)11-s + (−957. + 552. i)12-s − 3.05e3i·13-s + 2.97e3·15-s + (1.80e3 + 3.12e3i)16-s + (−2.46e3 − 1.42e3i)17-s + (−1.41e3 + 2.44e3i)18-s + (3.42e3 − 1.97e3i)19-s + ⋯
L(s)  = 1  + (0.725 + 1.25i)2-s + (0.5 + 0.288i)3-s + (−0.554 + 0.959i)4-s + (1.32 − 0.763i)5-s + 0.838i·6-s − 0.157·8-s + (0.166 + 0.288i)9-s + (1.92 + 1.10i)10-s + (0.772 − 1.33i)11-s + (−0.554 + 0.319i)12-s − 1.39i·13-s + 0.881·15-s + (0.439 + 0.762i)16-s + (−0.502 − 0.290i)17-s + (−0.241 + 0.419i)18-s + (0.498 − 0.287i)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

\(L(\frac{7}{2})\) \(\approx\) \(4.835542693\)
\(L(\frac12)\) \(\approx\) \(4.835542693\)
\(L(4)\) 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 \( 1 + (-5.80 - 10.0i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-165. + 95.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.02e3 + 1.77e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 3.05e3iT - 4.82e6T^{2} \)
17 \( 1 + (2.46e3 + 1.42e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-3.42e3 + 1.97e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (330. + 572. i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 9.28e3T + 5.94e8T^{2} \)
31 \( 1 + (2.42e3 + 1.40e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.84e4 - 3.19e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 6.79e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.23e4T + 6.32e9T^{2} \)
47 \( 1 + (1.27e5 - 7.38e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.09e5 - 1.90e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (1.66e5 + 9.62e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (2.88e5 - 1.66e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.74e5 - 3.02e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 3.05e5T + 1.28e11T^{2} \)
73 \( 1 + (-2.04e5 - 1.18e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.93e5 - 5.07e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 1.06e5iT - 3.26e11T^{2} \)
89 \( 1 + (1.13e4 - 6.52e3i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 2.05e5iT - 8.32e11T^{2} \)
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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

−12.73910139259843459941291241481, −10.99608456092826172086734895896, −9.716398269133725903527580009931, −8.783991316101865193722822061506, −7.85845951687826461885158850011, −6.33182776807453726404158783171, −5.63502985105594962034699008046, −4.67114547725669661560080299108, −3.11417774812553187781215481521, −1.20480951878724970909883903192, 1.82571799212543831045528562844, 1.99448868967633426148993840986, 3.50360561251473578879088099011, 4.70144996331582139283734495506, 6.31632472260759460187527972939, 7.26489842413710943036838781243, 9.290070406065486807080976505104, 9.787684528956455308249218914005, 10.87073234063935236804372544182, 11.90233786425231630807006808251

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