Properties

Label 2-471-1.1-c7-0-64
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  + 1.20·2-s − 27·3-s − 126.·4-s − 262.·5-s − 32.4·6-s + 1.19e3·7-s − 306.·8-s + 729·9-s − 315.·10-s + 3.81e3·11-s + 3.41e3·12-s + 5.19e3·13-s + 1.43e3·14-s + 7.08e3·15-s + 1.58e4·16-s + 3.55e4·17-s + 876.·18-s + 2.80e4·19-s + 3.32e4·20-s − 3.22e4·21-s + 4.58e3·22-s + 5.37e3·23-s + 8.26e3·24-s − 9.18e3·25-s + 6.25e3·26-s − 1.96e4·27-s − 1.51e5·28-s + ⋯
L(s)  = 1  + 0.106·2-s − 0.577·3-s − 0.988·4-s − 0.939·5-s − 0.0613·6-s + 1.31·7-s − 0.211·8-s + 0.333·9-s − 0.0998·10-s + 0.863·11-s + 0.570·12-s + 0.656·13-s + 0.140·14-s + 0.542·15-s + 0.966·16-s + 1.75·17-s + 0.0354·18-s + 0.937·19-s + 0.928·20-s − 0.760·21-s + 0.0917·22-s + 0.0920·23-s + 0.122·24-s − 0.117·25-s + 0.0697·26-s − 0.192·27-s − 1.30·28-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

\(L(4)\) \(\approx\) \(1.830459474\)
\(L(\frac12)\) \(\approx\) \(1.830459474\)
\(L(\frac{9}{2})\) 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 \( 1 - 1.20T + 128T^{2} \)
5 \( 1 + 262.T + 7.81e4T^{2} \)
7 \( 1 - 1.19e3T + 8.23e5T^{2} \)
11 \( 1 - 3.81e3T + 1.94e7T^{2} \)
13 \( 1 - 5.19e3T + 6.27e7T^{2} \)
17 \( 1 - 3.55e4T + 4.10e8T^{2} \)
19 \( 1 - 2.80e4T + 8.93e8T^{2} \)
23 \( 1 - 5.37e3T + 3.40e9T^{2} \)
29 \( 1 + 1.10e5T + 1.72e10T^{2} \)
31 \( 1 - 3.32e4T + 2.75e10T^{2} \)
37 \( 1 + 2.34e4T + 9.49e10T^{2} \)
41 \( 1 + 1.45e5T + 1.94e11T^{2} \)
43 \( 1 - 5.61e5T + 2.71e11T^{2} \)
47 \( 1 - 2.29e5T + 5.06e11T^{2} \)
53 \( 1 - 7.47e5T + 1.17e12T^{2} \)
59 \( 1 - 2.03e6T + 2.48e12T^{2} \)
61 \( 1 + 1.96e6T + 3.14e12T^{2} \)
67 \( 1 - 1.58e6T + 6.06e12T^{2} \)
71 \( 1 - 1.77e6T + 9.09e12T^{2} \)
73 \( 1 + 6.28e6T + 1.10e13T^{2} \)
79 \( 1 - 5.44e6T + 1.92e13T^{2} \)
83 \( 1 + 7.66e6T + 2.71e13T^{2} \)
89 \( 1 - 9.29e6T + 4.42e13T^{2} \)
97 \( 1 - 1.00e7T + 8.07e13T^{2} \)
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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

−9.860266087344746481253986774095, −8.887887806305073643288667208110, −7.977733129134880015878680883602, −7.41990151783753233864737875078, −5.83734592526124149905722839189, −5.13101069957941307924785291092, −4.14199922173602065440209615888, −3.50441223032403512700541344624, −1.38013203065727561657756961058, −0.72184385706028779940259732636, 0.72184385706028779940259732636, 1.38013203065727561657756961058, 3.50441223032403512700541344624, 4.14199922173602065440209615888, 5.13101069957941307924785291092, 5.83734592526124149905722839189, 7.41990151783753233864737875078, 7.977733129134880015878680883602, 8.887887806305073643288667208110, 9.860266087344746481253986774095

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