Properties

Label 2-471-1.1-c7-0-48
Degree $2$
Conductor $471$
Sign $1$
Analytic cond. $147.133$
Root an. cond. $12.1298$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.6·2-s − 27·3-s + 149.·4-s − 390.·5-s − 449.·6-s + 1.53e3·7-s + 349.·8-s + 729·9-s − 6.49e3·10-s − 8.45e3·11-s − 4.02e3·12-s − 3.45e3·13-s + 2.55e4·14-s + 1.05e4·15-s − 1.32e4·16-s + 1.88e4·17-s + 1.21e4·18-s − 2.01e4·19-s − 5.81e4·20-s − 4.14e4·21-s − 1.40e5·22-s + 8.83e4·23-s − 9.44e3·24-s + 7.41e4·25-s − 5.75e4·26-s − 1.96e4·27-s + 2.28e5·28-s + ⋯
L(s)  = 1  + 1.47·2-s − 0.577·3-s + 1.16·4-s − 1.39·5-s − 0.849·6-s + 1.69·7-s + 0.241·8-s + 0.333·9-s − 2.05·10-s − 1.91·11-s − 0.672·12-s − 0.436·13-s + 2.48·14-s + 0.805·15-s − 0.808·16-s + 0.928·17-s + 0.490·18-s − 0.674·19-s − 1.62·20-s − 0.975·21-s − 2.81·22-s + 1.51·23-s − 0.139·24-s + 0.948·25-s − 0.641·26-s − 0.192·27-s + 1.96·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $1$
Analytic conductor: \(147.133\)
Root analytic conductor: \(12.1298\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.700855338\)
\(L(\frac12)\) \(\approx\) \(2.700855338\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
157 \( 1 + 3.86e6T \)
good2 \( 1 - 16.6T + 128T^{2} \)
5 \( 1 + 390.T + 7.81e4T^{2} \)
7 \( 1 - 1.53e3T + 8.23e5T^{2} \)
11 \( 1 + 8.45e3T + 1.94e7T^{2} \)
13 \( 1 + 3.45e3T + 6.27e7T^{2} \)
17 \( 1 - 1.88e4T + 4.10e8T^{2} \)
19 \( 1 + 2.01e4T + 8.93e8T^{2} \)
23 \( 1 - 8.83e4T + 3.40e9T^{2} \)
29 \( 1 - 1.11e4T + 1.72e10T^{2} \)
31 \( 1 + 1.63e5T + 2.75e10T^{2} \)
37 \( 1 + 2.33e5T + 9.49e10T^{2} \)
41 \( 1 - 4.92e5T + 1.94e11T^{2} \)
43 \( 1 - 2.35e5T + 2.71e11T^{2} \)
47 \( 1 + 6.98e5T + 5.06e11T^{2} \)
53 \( 1 - 9.18e5T + 1.17e12T^{2} \)
59 \( 1 - 2.27e6T + 2.48e12T^{2} \)
61 \( 1 + 1.21e6T + 3.14e12T^{2} \)
67 \( 1 - 2.94e5T + 6.06e12T^{2} \)
71 \( 1 - 1.73e6T + 9.09e12T^{2} \)
73 \( 1 - 3.98e6T + 1.10e13T^{2} \)
79 \( 1 + 4.22e6T + 1.92e13T^{2} \)
83 \( 1 + 6.35e6T + 2.71e13T^{2} \)
89 \( 1 - 7.22e6T + 4.42e13T^{2} \)
97 \( 1 - 1.09e6T + 8.07e13T^{2} \)
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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

−10.47016423792945595537939105391, −8.616842772613614040878172492987, −7.71110115909506139671301999470, −7.20972725414362917155679040498, −5.57069446304972091169944118758, −5.02146348882735225195855271773, −4.45844209875442343748542238411, −3.35723782196741993613821570503, −2.20954705309320418165566880077, −0.58381741405573602443208211570, 0.58381741405573602443208211570, 2.20954705309320418165566880077, 3.35723782196741993613821570503, 4.45844209875442343748542238411, 5.02146348882735225195855271773, 5.57069446304972091169944118758, 7.20972725414362917155679040498, 7.71110115909506139671301999470, 8.616842772613614040878172492987, 10.47016423792945595537939105391

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