Properties

Label 2-147-7.6-c6-0-28
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  − 13.1·2-s − 15.5i·3-s + 109.·4-s − 79.5i·5-s + 205. i·6-s − 601.·8-s − 243·9-s + 1.04e3i·10-s + 822.·11-s − 1.70e3i·12-s − 2.42e3i·13-s − 1.24e3·15-s + 913.·16-s − 7.79e3i·17-s + 3.20e3·18-s + 6.68e3i·19-s + ⋯
L(s)  = 1  − 1.64·2-s − 0.577i·3-s + 1.71·4-s − 0.636i·5-s + 0.951i·6-s − 1.17·8-s − 0.333·9-s + 1.04i·10-s + 0.617·11-s − 0.989i·12-s − 1.10i·13-s − 0.367·15-s + 0.223·16-s − 1.58i·17-s + 0.549·18-s + 0.974i·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

\(L(\frac{7}{2})\) \(\approx\) \(0.8188133939\)
\(L(\frac12)\) \(\approx\) \(0.8188133939\)
\(L(4)\) 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 \( 1 + 13.1T + 64T^{2} \)
5 \( 1 + 79.5iT - 1.56e4T^{2} \)
11 \( 1 - 822.T + 1.77e6T^{2} \)
13 \( 1 + 2.42e3iT - 4.82e6T^{2} \)
17 \( 1 + 7.79e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.68e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.88e4T + 1.48e8T^{2} \)
29 \( 1 - 1.38e4T + 5.94e8T^{2} \)
31 \( 1 - 2.78e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.96e4T + 2.56e9T^{2} \)
41 \( 1 + 5.91e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.18e4T + 6.32e9T^{2} \)
47 \( 1 + 5.01e3iT - 1.07e10T^{2} \)
53 \( 1 - 1.86e5T + 2.21e10T^{2} \)
59 \( 1 + 2.25e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.44e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.35e5T + 9.04e10T^{2} \)
71 \( 1 - 9.62e4T + 1.28e11T^{2} \)
73 \( 1 + 2.75e5iT - 1.51e11T^{2} \)
79 \( 1 + 6.81e5T + 2.43e11T^{2} \)
83 \( 1 - 1.28e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.72e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.20e5iT - 8.32e11T^{2} \)
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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

−11.38731960545699456357988652706, −10.31956488114792054417417528968, −9.270678684099442513311577338819, −8.537182120657453174974005233030, −7.57599449978181353800684500579, −6.66540427706292542569006753684, −5.11580960274797517862985327849, −2.86835903143948128891538171692, −1.27437456335060212881789862430, −0.54221659349303010969648061742, 1.18780453035651825346332890748, 2.67194511566084456211540819502, 4.33567592810404601831685662055, 6.34053773972021615641053671251, 7.13385324657553115067452288922, 8.501657367656905142092228460726, 9.187350098992822755628336890492, 10.12164970197107064672563149925, 11.01797942526577519045005852243, 11.59656378146180161437561671056

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