Properties

Label 2-471-1.1-c7-0-91
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  + 6.39·2-s − 27·3-s − 87.1·4-s + 330.·5-s − 172.·6-s + 172.·7-s − 1.37e3·8-s + 729·9-s + 2.11e3·10-s + 7.16e3·11-s + 2.35e3·12-s + 1.44e4·13-s + 1.10e3·14-s − 8.92e3·15-s + 2.35e3·16-s + 1.34e4·17-s + 4.66e3·18-s + 4.63e4·19-s − 2.88e4·20-s − 4.66e3·21-s + 4.58e4·22-s − 6.70e4·23-s + 3.71e4·24-s + 3.11e4·25-s + 9.21e4·26-s − 1.96e4·27-s − 1.50e4·28-s + ⋯
L(s)  = 1  + 0.565·2-s − 0.577·3-s − 0.680·4-s + 1.18·5-s − 0.326·6-s + 0.190·7-s − 0.949·8-s + 0.333·9-s + 0.668·10-s + 1.62·11-s + 0.392·12-s + 1.81·13-s + 0.107·14-s − 0.682·15-s + 0.143·16-s + 0.662·17-s + 0.188·18-s + 1.55·19-s − 0.805·20-s − 0.109·21-s + 0.917·22-s − 1.14·23-s + 0.548·24-s + 0.398·25-s + 1.02·26-s − 0.192·27-s − 0.129·28-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

\(L(4)\) \(\approx\) \(3.618190605\)
\(L(\frac12)\) \(\approx\) \(3.618190605\)
\(L(\frac{9}{2})\) 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 \( 1 - 6.39T + 128T^{2} \)
5 \( 1 - 330.T + 7.81e4T^{2} \)
7 \( 1 - 172.T + 8.23e5T^{2} \)
11 \( 1 - 7.16e3T + 1.94e7T^{2} \)
13 \( 1 - 1.44e4T + 6.27e7T^{2} \)
17 \( 1 - 1.34e4T + 4.10e8T^{2} \)
19 \( 1 - 4.63e4T + 8.93e8T^{2} \)
23 \( 1 + 6.70e4T + 3.40e9T^{2} \)
29 \( 1 - 9.37e4T + 1.72e10T^{2} \)
31 \( 1 + 8.70e4T + 2.75e10T^{2} \)
37 \( 1 + 4.73e5T + 9.49e10T^{2} \)
41 \( 1 - 6.84e5T + 1.94e11T^{2} \)
43 \( 1 + 2.57e5T + 2.71e11T^{2} \)
47 \( 1 - 1.13e6T + 5.06e11T^{2} \)
53 \( 1 + 1.41e6T + 1.17e12T^{2} \)
59 \( 1 + 4.76e5T + 2.48e12T^{2} \)
61 \( 1 - 2.14e6T + 3.14e12T^{2} \)
67 \( 1 - 3.05e5T + 6.06e12T^{2} \)
71 \( 1 + 2.57e6T + 9.09e12T^{2} \)
73 \( 1 - 2.90e6T + 1.10e13T^{2} \)
79 \( 1 - 5.80e6T + 1.92e13T^{2} \)
83 \( 1 - 5.38e6T + 2.71e13T^{2} \)
89 \( 1 + 9.71e6T + 4.42e13T^{2} \)
97 \( 1 + 1.09e7T + 8.07e13T^{2} \)
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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

−9.670141160340283408179934950886, −9.251644380614899038348350888784, −8.203844313472919266511559889152, −6.65226741399794856442989637361, −5.90747177171357955633625074120, −5.41389846319551672195932395929, −4.12414446973280762459983014869, −3.38935679515821007707398546486, −1.57598176797256848206668495355, −0.915289998714314414192483397279, 0.915289998714314414192483397279, 1.57598176797256848206668495355, 3.38935679515821007707398546486, 4.12414446973280762459983014869, 5.41389846319551672195932395929, 5.90747177171357955633625074120, 6.65226741399794856442989637361, 8.203844313472919266511559889152, 9.251644380614899038348350888784, 9.670141160340283408179934950886

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