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Properties

Label 2-147-21.20-c5-0-21
Degree $2$
Conductor $147$
Sign $-0.997 - 0.0755i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.3i·2-s + (12.5 − 9.28i)3-s − 74.8·4-s + 31.7·5-s + (95.9 + 129. i)6-s − 442. i·8-s + (70.5 − 232. i)9-s + 328. i·10-s + 453. i·11-s + (−936. + 694. i)12-s + 551. i·13-s + (397. − 294. i)15-s + 2.17e3·16-s + 538.·17-s + (2.40e3 + 729. i)18-s + 1.36e3i·19-s + ⋯
L(s)  = 1  + 1.82i·2-s + (0.803 − 0.595i)3-s − 2.33·4-s + 0.567·5-s + (1.08 + 1.46i)6-s − 2.44i·8-s + (0.290 − 0.956i)9-s + 1.03i·10-s + 1.13i·11-s + (−1.87 + 1.39i)12-s + 0.905i·13-s + (0.456 − 0.338i)15-s + 2.12·16-s + 0.451·17-s + (1.74 + 0.530i)18-s + 0.868i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0755i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.997 - 0.0755i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.997 - 0.0755i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.065256194\)
\(L(\frac12)\) \(\approx\) \(2.065256194\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-12.5 + 9.28i)T \)
7 \( 1 \)
good2 \( 1 - 10.3iT - 32T^{2} \)
5 \( 1 - 31.7T + 3.12e3T^{2} \)
11 \( 1 - 453. iT - 1.61e5T^{2} \)
13 \( 1 - 551. iT - 3.71e5T^{2} \)
17 \( 1 - 538.T + 1.41e6T^{2} \)
19 \( 1 - 1.36e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.23e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.60e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.06e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.95e3T + 6.93e7T^{2} \)
41 \( 1 - 1.80e3T + 1.15e8T^{2} \)
43 \( 1 - 7.88e3T + 1.47e8T^{2} \)
47 \( 1 + 6.09e3T + 2.29e8T^{2} \)
53 \( 1 + 1.41e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.69e4T + 7.14e8T^{2} \)
61 \( 1 + 2.97e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.36e4T + 1.35e9T^{2} \)
71 \( 1 - 3.13e4iT - 1.80e9T^{2} \)
73 \( 1 + 9.82e3iT - 2.07e9T^{2} \)
79 \( 1 - 9.97e4T + 3.07e9T^{2} \)
83 \( 1 + 7.95e4T + 3.93e9T^{2} \)
89 \( 1 - 953.T + 5.58e9T^{2} \)
97 \( 1 - 1.15e5iT - 8.58e9T^{2} \)
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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

−13.11242847235928724735000457446, −12.14422388142696222982511136741, −9.840315142216608631546996718557, −9.261714412127142254704397946972, −8.130111738427793388703526316662, −7.27067183446692306916048016564, −6.49348034959658641077974256776, −5.29563066432323496730273193861, −3.86560714547947192078378353799, −1.67259314751575459525657570244, 0.65387480604710698956255772656, 2.29843476779164296233528912695, 3.14265616060194153386868106795, 4.30258598551283273855965262265, 5.61818714256207048403805769753, 8.066264798800074743763607551732, 8.962405921018631973178574005881, 9.837205170684750236540164313221, 10.58734574080317143693081177937, 11.38169691268023193830332433379

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