Properties

Degree $2$
Conductor $147$
Sign $-0.0725 + 0.997i$
Motivic weight $1$
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 + (0.5 + 0.866i)3-s + (−1.91 − 3.31i)4-s + (0.292 − 0.507i)5-s + 2.41·6-s − 4.41·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (1 + 1.73i)11-s + (1.91 − 3.31i)12-s − 5.41·13-s + 0.585·15-s + (−1.49 + 2.59i)16-s + (3.12 + 5.40i)17-s + (1.20 + 2.09i)18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (0.853 − 1.47i)2-s + (0.288 + 0.499i)3-s + (−0.957 − 1.65i)4-s + (0.130 − 0.226i)5-s + 0.985·6-s − 1.56·8-s + (−0.166 + 0.288i)9-s + (−0.223 − 0.387i)10-s + (0.301 + 0.522i)11-s + (0.552 − 0.957i)12-s − 1.50·13-s + 0.151·15-s + (−0.374 + 0.649i)16-s + (0.757 + 1.31i)17-s + (0.284 + 0.492i)18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.0725 + 0.997i$
Motivic weight: \(1\)
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15995 - 1.24740i\)
\(L(\frac12)\) \(\approx\) \(1.15995 - 1.24740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.82 - 3.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (3.41 + 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.41 + 5.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.87 + 3.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + (2.94 + 5.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + (2.87 - 4.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.41T + 97T^{2} \)
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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

−12.66702220215125935320562188260, −11.92217519573507199466513811231, −10.87811590835846789043656763950, −9.889510928048099504963370216341, −9.311656244740961829791763617556, −7.57433717323487988304552963124, −5.57176317418056938817905304053, −4.57043138743411482840729084786, −3.43581297509463027675472343068, −1.98677109128099668675269595975, 3.03684330363718662259261721206, 4.69307314737029500098423310459, 5.80647170531078770796941100252, 6.94583933269027828633817373627, 7.61294315908616873244959779219, 8.727657081669744181563898226769, 10.04692161610703631011305794187, 11.87814090679738796004096596894, 12.58572258318403150468513277789, 13.75230583028398820210289807479

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