Properties

Label 2-471-1.1-c7-0-1
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  − 19.9·2-s − 27·3-s + 271.·4-s − 231.·5-s + 539.·6-s + 342.·7-s − 2.87e3·8-s + 729·9-s + 4.63e3·10-s − 1.17e3·11-s − 7.33e3·12-s − 8.82e3·13-s − 6.84e3·14-s + 6.26e3·15-s + 2.27e4·16-s − 924.·17-s − 1.45e4·18-s − 4.18e4·19-s − 6.30e4·20-s − 9.24e3·21-s + 2.35e4·22-s + 3.63e4·23-s + 7.76e4·24-s − 2.43e4·25-s + 1.76e5·26-s − 1.96e4·27-s + 9.30e4·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.577·3-s + 2.12·4-s − 0.829·5-s + 1.02·6-s + 0.377·7-s − 1.98·8-s + 0.333·9-s + 1.46·10-s − 0.267·11-s − 1.22·12-s − 1.11·13-s − 0.666·14-s + 0.478·15-s + 1.38·16-s − 0.0456·17-s − 0.589·18-s − 1.39·19-s − 1.76·20-s − 0.217·21-s + 0.472·22-s + 0.623·23-s + 1.14·24-s − 0.311·25-s + 1.96·26-s − 0.192·27-s + 0.801·28-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

\(L(4)\) \(\approx\) \(0.01740786022\)
\(L(\frac12)\) \(\approx\) \(0.01740786022\)
\(L(\frac{9}{2})\) 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 \( 1 + 19.9T + 128T^{2} \)
5 \( 1 + 231.T + 7.81e4T^{2} \)
7 \( 1 - 342.T + 8.23e5T^{2} \)
11 \( 1 + 1.17e3T + 1.94e7T^{2} \)
13 \( 1 + 8.82e3T + 6.27e7T^{2} \)
17 \( 1 + 924.T + 4.10e8T^{2} \)
19 \( 1 + 4.18e4T + 8.93e8T^{2} \)
23 \( 1 - 3.63e4T + 3.40e9T^{2} \)
29 \( 1 + 1.44e5T + 1.72e10T^{2} \)
31 \( 1 + 9.00e4T + 2.75e10T^{2} \)
37 \( 1 + 5.50e5T + 9.49e10T^{2} \)
41 \( 1 - 5.09e5T + 1.94e11T^{2} \)
43 \( 1 - 5.35e4T + 2.71e11T^{2} \)
47 \( 1 - 1.21e6T + 5.06e11T^{2} \)
53 \( 1 + 1.60e6T + 1.17e12T^{2} \)
59 \( 1 - 6.29e4T + 2.48e12T^{2} \)
61 \( 1 + 1.02e6T + 3.14e12T^{2} \)
67 \( 1 - 2.39e6T + 6.06e12T^{2} \)
71 \( 1 - 3.52e6T + 9.09e12T^{2} \)
73 \( 1 - 9.15e5T + 1.10e13T^{2} \)
79 \( 1 + 3.18e6T + 1.92e13T^{2} \)
83 \( 1 + 5.91e6T + 2.71e13T^{2} \)
89 \( 1 + 1.32e7T + 4.42e13T^{2} \)
97 \( 1 + 8.60e6T + 8.07e13T^{2} \)
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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

−9.806364026104557501304639183666, −8.979783116121220919129198763791, −8.045327340581384137352101300410, −7.41646046947707185359324311059, −6.67488662768900147507804831155, −5.35585711897581910129071390783, −4.09840353642704883314249490530, −2.49630085362367100992703452085, −1.48632029922744923258856933768, −0.080266627118168773441669969947, 0.080266627118168773441669969947, 1.48632029922744923258856933768, 2.49630085362367100992703452085, 4.09840353642704883314249490530, 5.35585711897581910129071390783, 6.67488662768900147507804831155, 7.41646046947707185359324311059, 8.045327340581384137352101300410, 8.979783116121220919129198763791, 9.806364026104557501304639183666

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