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Properties

Label 2-147-7.5-c6-0-26
Degree $2$
Conductor $147$
Sign $0.553 + 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.79 + 13.5i)2-s + (13.5 − 7.79i)3-s + (−89.6 − 155. i)4-s + (−22.3 − 12.9i)5-s + 243. i·6-s + 1.79e3·8-s + (121.5 − 210. i)9-s + (348. − 201. i)10-s + (−311. − 540. i)11-s + (−2.41e3 − 1.39e3i)12-s + 3.25e3i·13-s − 402.·15-s + (−8.27e3 + 1.43e4i)16-s + (−275. + 158. i)17-s + (1.89e3 + 3.28e3i)18-s + (5.19e3 + 3.00e3i)19-s + ⋯
L(s)  = 1  + (−0.974 + 1.68i)2-s + (0.5 − 0.288i)3-s + (−1.40 − 2.42i)4-s + (−0.178 − 0.103i)5-s + 1.12i·6-s + 3.50·8-s + (0.166 − 0.288i)9-s + (0.348 − 0.201i)10-s + (−0.234 − 0.405i)11-s + (−1.40 − 0.808i)12-s + 1.48i·13-s − 0.119·15-s + (−2.02 + 3.49i)16-s + (−0.0560 + 0.0323i)17-s + (0.324 + 0.562i)18-s + (0.757 + 0.437i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4000430745\)
\(L(\frac12)\) \(\approx\) \(0.4000430745\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (7.79 - 13.5i)T + (-32 - 55.4i)T^{2} \)
5 \( 1 + (22.3 + 12.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (311. + 540. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 3.25e3iT - 4.82e6T^{2} \)
17 \( 1 + (275. - 158. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.19e3 - 3.00e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-21.0 + 36.3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 2.42e4T + 5.94e8T^{2} \)
31 \( 1 + (-1.75e4 + 1.01e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-8.68e3 + 1.50e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 1.00e5iT - 4.75e9T^{2} \)
43 \( 1 + 6.78e4T + 6.32e9T^{2} \)
47 \( 1 + (5.70e4 + 3.29e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (4.96e4 + 8.59e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (8.71e4 - 5.03e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-2.01e5 - 1.16e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.05e4 + 3.55e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 4.00e5T + 1.28e11T^{2} \)
73 \( 1 + (4.79e5 - 2.76e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-5.24e3 + 9.07e3i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 7.12e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.64e5 - 9.52e4i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.07e6iT - 8.32e11T^{2} \)
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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

−11.63314655736396598827520185649, −10.18162279373501749431710497120, −9.267880734974254689295351583442, −8.481394259109586747305552602146, −7.57266026609390598637610843794, −6.70047255968686075179198247010, −5.62632220432911732903051011526, −4.21656101287895391564004674142, −1.67303280729078557028447042567, −0.17152523095953566220513957393, 1.28737822696663323301445944971, 2.72363926594664791243632866602, 3.50981227811407635977942529557, 4.91528117441195955762495547011, 7.52502113485251122697024755417, 8.193533325857616622544128247961, 9.364192378779078918518841493980, 10.03411598652901465738736392472, 10.93090057229470683313371887295, 11.80521722124247557270292105049

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