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Properties

Label 2-147-7.4-c5-0-17
Degree $2$
Conductor $147$
Sign $-0.749 - 0.661i$
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  + (−5.00 − 8.66i)2-s + (−4.5 + 7.79i)3-s + (−34.0 + 59.0i)4-s + (−35.1 − 60.9i)5-s + 90.0·6-s + 362.·8-s + (−40.5 − 70.1i)9-s + (−352. + 609. i)10-s + (365. − 633. i)11-s + (−306. − 531. i)12-s + 899.·13-s + 633.·15-s + (−721. − 1.24e3i)16-s + (696. − 1.20e3i)17-s + (−405. + 702. i)18-s + (95.0 + 164. i)19-s + ⋯
L(s)  = 1  + (−0.884 − 1.53i)2-s + (−0.288 + 0.499i)3-s + (−1.06 + 1.84i)4-s + (−0.629 − 1.08i)5-s + 1.02·6-s + 2.00·8-s + (−0.166 − 0.288i)9-s + (−1.11 + 1.92i)10-s + (0.911 − 1.57i)11-s + (−0.615 − 1.06i)12-s + 1.47·13-s + 0.726·15-s + (−0.704 − 1.21i)16-s + (0.584 − 1.01i)17-s + (−0.294 + 0.510i)18-s + (0.0604 + 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7106557452\)
\(L(\frac12)\) \(\approx\) \(0.7106557452\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (5.00 + 8.66i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (35.1 + 60.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-365. + 633. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 899.T + 3.71e5T^{2} \)
17 \( 1 + (-696. + 1.20e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-95.0 - 164. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (21.4 + 37.2i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 7.74e3T + 2.05e7T^{2} \)
31 \( 1 + (589. - 1.02e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.64e3 + 8.04e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.34e4T + 1.15e8T^{2} \)
43 \( 1 - 6.03e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.62e3 - 2.81e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.28e4 - 2.22e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.32e4 + 2.29e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-3.21e3 - 5.57e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.19e4 + 2.06e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.46e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.98e4 + 3.42e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.15e4 + 2.00e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.72e4T + 3.93e9T^{2} \)
89 \( 1 + (2.49e4 + 4.32e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.68e4T + 8.58e9T^{2} \)
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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

−11.32197589563775664742970987312, −10.82856862295211263643731790051, −9.236477808260204809648889096368, −8.958905414731380516650635976283, −7.929870572065667285196055412223, −5.74854252571261259200796921800, −4.07977581372229727662533161048, −3.36526867780575359864529230216, −1.20902909931353525503146644939, −0.43165391831811401161443841357, 1.42976967365618028285255605348, 3.94602587885817308680901573884, 5.75764033262259911647366741268, 6.69620090924185040327885292964, 7.31149747762111599532943981673, 8.228999261516952209622050925049, 9.426091751961820216435252162829, 10.52928027472835173532504096356, 11.54471521051132742648141134344, 12.90989139939310023905100754672

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