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Properties

Label 2-147-21.20-c3-0-14
Degree $2$
Conductor $147$
Sign $-0.827 - 0.561i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.84i·2-s + (5.11 + 0.905i)3-s − 15.4·4-s + 17.3·5-s + (−4.38 + 24.7i)6-s − 36.2i·8-s + (25.3 + 9.26i)9-s + 83.8i·10-s + 27.8i·11-s + (−79.1 − 14.0i)12-s − 16.3i·13-s + (88.5 + 15.6i)15-s + 51.6·16-s − 40.8·17-s + (−44.8 + 122. i)18-s + 68.1i·19-s + ⋯
L(s)  = 1  + 1.71i·2-s + (0.984 + 0.174i)3-s − 1.93·4-s + 1.54·5-s + (−0.298 + 1.68i)6-s − 1.59i·8-s + (0.939 + 0.343i)9-s + 2.65i·10-s + 0.763i·11-s + (−1.90 − 0.336i)12-s − 0.347i·13-s + (1.52 + 0.269i)15-s + 0.806·16-s − 0.582·17-s + (−0.587 + 1.60i)18-s + 0.823i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.763492 + 2.48336i\)
\(L(\frac12)\) \(\approx\) \(0.763492 + 2.48336i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.11 - 0.905i)T \)
7 \( 1 \)
good2 \( 1 - 4.84iT - 8T^{2} \)
5 \( 1 - 17.3T + 125T^{2} \)
11 \( 1 - 27.8iT - 1.33e3T^{2} \)
13 \( 1 + 16.3iT - 2.19e3T^{2} \)
17 \( 1 + 40.8T + 4.91e3T^{2} \)
19 \( 1 - 68.1iT - 6.85e3T^{2} \)
23 \( 1 + 71.7iT - 1.21e4T^{2} \)
29 \( 1 + 216. iT - 2.43e4T^{2} \)
31 \( 1 + 157. iT - 2.97e4T^{2} \)
37 \( 1 + 348.T + 5.06e4T^{2} \)
41 \( 1 + 153.T + 6.89e4T^{2} \)
43 \( 1 - 427.T + 7.95e4T^{2} \)
47 \( 1 + 16.8T + 1.03e5T^{2} \)
53 \( 1 - 192. iT - 1.48e5T^{2} \)
59 \( 1 - 287.T + 2.05e5T^{2} \)
61 \( 1 - 224. iT - 2.26e5T^{2} \)
67 \( 1 + 172.T + 3.00e5T^{2} \)
71 \( 1 + 1.06e3iT - 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 52.0T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + 668.T + 7.04e5T^{2} \)
97 \( 1 + 1.35e3iT - 9.12e5T^{2} \)
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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

−13.56628006368513657398771318112, −12.67990740185265604945609688013, −10.28774018880482782554399250513, −9.556019315030234246912840827289, −8.708115272064334107222496976544, −7.66154545431678876606364024736, −6.58466866723482723088147960987, −5.57918157073915354683610812134, −4.34372871553265762996614775497, −2.18196595613853819411670635355, 1.37299977745974904112202751268, 2.38187371162197592438295660833, 3.46654223101668622259960147770, 5.07718103690872557044471667732, 6.78016467006749093464168922607, 8.789111007117632635306274893177, 9.174973613522885168457076654488, 10.18578314406675410786135818299, 10.97382640264943493066861383011, 12.32651606992794350202072896533

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