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Properties

Label 2-147-7.4-c3-0-2
Degree $2$
Conductor $147$
Sign $-0.0725 - 0.997i$
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  + (−1.20 − 2.09i)2-s + (−1.5 + 2.59i)3-s + (1.08 − 1.88i)4-s + (9.94 + 17.2i)5-s + 7.24·6-s − 24.5·8-s + (−4.5 − 7.79i)9-s + (24.0 − 41.6i)10-s + (−11.9 + 20.7i)11-s + (3.25 + 5.64i)12-s − 87.3·13-s − 59.6·15-s + (20.9 + 36.2i)16-s + (−2.81 + 4.88i)17-s + (−10.8 + 18.8i)18-s + (32.4 + 56.1i)19-s + ⋯
L(s)  = 1  + (−0.426 − 0.739i)2-s + (−0.288 + 0.499i)3-s + (0.135 − 0.235i)4-s + (0.889 + 1.54i)5-s + 0.492·6-s − 1.08·8-s + (−0.166 − 0.288i)9-s + (0.759 − 1.31i)10-s + (−0.328 + 0.568i)11-s + (0.0783 + 0.135i)12-s − 1.86·13-s − 1.02·15-s + (0.327 + 0.567i)16-s + (−0.0402 + 0.0696i)17-s + (−0.142 + 0.246i)18-s + (0.391 + 0.678i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.0725 - 0.997i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.583936 + 0.627964i\)
\(L(\frac12)\) \(\approx\) \(0.583936 + 0.627964i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (1.20 + 2.09i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-9.94 - 17.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (11.9 - 20.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 87.3T + 2.19e3T^{2} \)
17 \( 1 + (2.81 - 4.88i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-32.4 - 56.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-12.7 - 22.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 60.3T + 2.43e4T^{2} \)
31 \( 1 + (61.3 - 106. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-28.0 - 48.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 299.T + 6.89e4T^{2} \)
43 \( 1 + 501.T + 7.95e4T^{2} \)
47 \( 1 + (-152. - 264. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-187. + 324. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (313. - 543. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-1.87 - 3.25i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-406. + 704. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 165.T + 3.57e5T^{2} \)
73 \( 1 + (-309. + 536. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-69.1 - 119. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 621.T + 5.71e5T^{2} \)
89 \( 1 + (-142. - 247. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 603.T + 9.12e5T^{2} \)
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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

−12.49783601024574347448955534572, −11.54518522835499532836888572510, −10.52809398826320600301741982975, −10.03953602284599750670186949736, −9.429276254794841221700574429683, −7.38571021235192491930614930907, −6.35043217765508543467318021001, −5.19217255359235639264689126725, −3.08265946763753368443205584411, −2.09312749412864762960085492536, 0.45693484298887249970441401626, 2.41448032202452045622379673752, 4.91582320142464900671091672041, 5.77772507154076857588648576320, 7.05317882462747568893306197390, 8.083271500471040853015214583793, 9.017627183103384361066038825730, 9.869508318529475953366075184358, 11.62347091490017075505991604898, 12.48803091011496748746959326531

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