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Properties

Label 2-147-7.6-c6-0-38
Degree $2$
Conductor $147$
Sign $-0.654 + 0.755i$
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  + 12.4·2-s − 15.5i·3-s + 91.6·4-s − 202. i·5-s − 194. i·6-s + 345.·8-s − 243·9-s − 2.52e3i·10-s + 875.·11-s − 1.42e3i·12-s − 275. i·13-s − 3.15e3·15-s − 1.56e3·16-s + 4.38e3i·17-s − 3.03e3·18-s − 1.35e4i·19-s + ⋯
L(s)  = 1  + 1.55·2-s − 0.577i·3-s + 1.43·4-s − 1.61i·5-s − 0.900i·6-s + 0.674·8-s − 0.333·9-s − 2.52i·10-s + 0.657·11-s − 0.826i·12-s − 0.125i·13-s − 0.935·15-s − 0.380·16-s + 0.892i·17-s − 0.519·18-s − 1.96i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ -0.654 + 0.755i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.213219077\)
\(L(\frac12)\) \(\approx\) \(4.213219077\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5iT \)
7 \( 1 \)
good2 \( 1 - 12.4T + 64T^{2} \)
5 \( 1 + 202. iT - 1.56e4T^{2} \)
11 \( 1 - 875.T + 1.77e6T^{2} \)
13 \( 1 + 275. iT - 4.82e6T^{2} \)
17 \( 1 - 4.38e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.35e4iT - 4.70e7T^{2} \)
23 \( 1 + 1.27e4T + 1.48e8T^{2} \)
29 \( 1 - 6.26e3T + 5.94e8T^{2} \)
31 \( 1 - 2.04e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.57e4T + 2.56e9T^{2} \)
41 \( 1 + 6.99e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.13e5T + 6.32e9T^{2} \)
47 \( 1 + 4.63e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.28e5T + 2.21e10T^{2} \)
59 \( 1 + 2.80e5iT - 4.21e10T^{2} \)
61 \( 1 - 9.78e4iT - 5.15e10T^{2} \)
67 \( 1 - 1.74e5T + 9.04e10T^{2} \)
71 \( 1 - 3.45e5T + 1.28e11T^{2} \)
73 \( 1 - 1.20e5iT - 1.51e11T^{2} \)
79 \( 1 - 7.63e5T + 2.43e11T^{2} \)
83 \( 1 - 8.59e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.79e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.40e5iT - 8.32e11T^{2} \)
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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

−12.17755473050422857801899591655, −11.12665386042442866817423317578, −9.270781765360134149079088806270, −8.382827982182927536144113933386, −6.87940609459416690444611895948, −5.77317824668474811932465007616, −4.83416744569482972080729885429, −3.87527670076996405496619331026, −2.16671173265970664414758182462, −0.71965927962071644149456285154, 2.35192085183571367859660229850, 3.44564574445009132289307818736, 4.21420277282316136860174770129, 5.78073237339797732807104223025, 6.46415458492552600255182320000, 7.68772899108811518664504562891, 9.570886058087767035875028582386, 10.56350486333223669159203386443, 11.53071624740371882428381020716, 12.18233639852026233489260658961

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