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Properties

Label 2-147-21.17-c3-0-4
Degree $2$
Conductor $147$
Sign $-0.929 - 0.368i$
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 + (−1.60 + 4.94i)3-s + (−2.17 + 3.77i)4-s + (0.623 + 1.08i)5-s + (2.05 + 9.70i)6-s + 23.5i·8-s + (−21.8 − 15.8i)9-s + (2.06 + 1.19i)10-s + (−35.2 − 20.3i)11-s + (−15.1 − 16.8i)12-s − 19.5i·13-s + (−6.34 + 1.34i)15-s + (5.08 + 8.80i)16-s + (−52.3 + 90.6i)17-s + (−51.2 − 5.42i)18-s + (−35.0 + 20.2i)19-s + ⋯
L(s)  = 1  + (0.584 − 0.337i)2-s + (−0.309 + 0.950i)3-s + (−0.272 + 0.471i)4-s + (0.0557 + 0.0966i)5-s + (0.140 + 0.660i)6-s + 1.04i·8-s + (−0.808 − 0.588i)9-s + (0.0652 + 0.0376i)10-s + (−0.965 − 0.557i)11-s + (−0.364 − 0.404i)12-s − 0.418i·13-s + (−0.109 + 0.0231i)15-s + (0.0794 + 0.137i)16-s + (−0.746 + 1.29i)17-s + (−0.671 − 0.0710i)18-s + (−0.423 + 0.244i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.929 - 0.368i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.181054 + 0.946822i\)
\(L(\frac12)\) \(\approx\) \(0.181054 + 0.946822i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 - 4.94i)T \)
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

−12.93809698552998138451176647188, −12.10104649305902494158820369694, −10.90012514502808185302105128116, −10.37366490599499480749141070876, −8.828307106969058296603087582500, −8.086943918382458914156675590499, −6.13779349535528974926121722870, −5.03325823327685562903058981536, −3.95753263662217734848498012137, −2.79516530971974862537371183941, 0.38732010565137903778670395996, 2.30554378272216729502854269253, 4.52588235588002682314536877196, 5.51921873661192181259151175517, 6.62986628264734825602373977728, 7.53176469438239972156118641819, 8.958335775329536693371838273327, 10.15870503428330613331626013450, 11.33830327406688333819199237393, 12.42054158504908828205887358640

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