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Properties

Label 2-147-21.5-c3-0-12
Degree $2$
Conductor $147$
Sign $0.699 - 0.714i$
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  + (−1.65 − 0.954i)2-s + (3.47 + 3.86i)3-s + (−2.17 − 3.77i)4-s + (−0.623 + 1.08i)5-s + (−2.05 − 9.70i)6-s + 23.5i·8-s + (−2.84 + 26.8i)9-s + (2.06 − 1.19i)10-s + (35.2 − 20.3i)11-s + (7.00 − 21.5i)12-s + 19.5i·13-s + (−6.34 + 1.34i)15-s + (5.08 − 8.80i)16-s + (52.3 + 90.6i)17-s + (30.3 − 41.6i)18-s + (−35.0 − 20.2i)19-s + ⋯
L(s)  = 1  + (−0.584 − 0.337i)2-s + (0.668 + 0.743i)3-s + (−0.272 − 0.471i)4-s + (−0.0557 + 0.0966i)5-s + (−0.140 − 0.660i)6-s + 1.04i·8-s + (−0.105 + 0.994i)9-s + (0.0652 − 0.0376i)10-s + (0.965 − 0.557i)11-s + (0.168 − 0.517i)12-s + 0.418i·13-s + (−0.109 + 0.0231i)15-s + (0.0794 − 0.137i)16-s + (0.746 + 1.29i)17-s + (0.397 − 0.545i)18-s + (−0.423 − 0.244i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.699 - 0.714i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.25633 + 0.528255i\)
\(L(\frac12)\) \(\approx\) \(1.25633 + 0.528255i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.47 - 3.86i)T \)
7 \( 1 \)
good2 \( 1 + (1.65 + 0.954i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (0.623 - 1.08i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.2 + 20.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 19.5iT - 2.19e3T^{2} \)
17 \( 1 + (-52.3 - 90.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (35.0 + 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-69.6 - 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 211. iT - 2.43e4T^{2} \)
31 \( 1 + (-86.6 + 50.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-94.9 + 164. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 186.T + 6.89e4T^{2} \)
43 \( 1 - 158.T + 7.95e4T^{2} \)
47 \( 1 + (179. - 310. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (366. - 211. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (312. + 541. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (699. + 403. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (149. + 258. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 455. iT - 3.57e5T^{2} \)
73 \( 1 + (-434. + 250. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-30.9 + 53.6i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 73.1T + 5.71e5T^{2} \)
89 \( 1 + (57.3 - 99.3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.41e3iT - 9.12e5T^{2} \)
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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

−12.74032675333733940716244655783, −11.15174411162624050590051855718, −10.68995899185247433730231490477, −9.403725832638647210254585689119, −8.980037262522471449789994675689, −7.85640629522851430355310938619, −6.09128466047077092793333507578, −4.70961596908402296159441419654, −3.35167524234405442744376686710, −1.53126271009434502182557264947, 0.852026451874840085200361466323, 2.89166843977737905920188967364, 4.33740872194902129779493790412, 6.41441063836415346269212382602, 7.35322021683539532655192320147, 8.210625560760849984886796927809, 9.143487450080571862084075067789, 9.949849184755118201559318370150, 11.82386867755028336094301660060, 12.46095063820371374899346040132

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