Properties

Label 2-471-1.1-c7-0-35
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  − 12.5·2-s − 27·3-s + 29.7·4-s + 553.·5-s + 339.·6-s − 406.·7-s + 1.23e3·8-s + 729·9-s − 6.95e3·10-s + 2.65e3·11-s − 802.·12-s − 1.48e4·13-s + 5.10e3·14-s − 1.49e4·15-s − 1.93e4·16-s − 2.19e4·17-s − 9.15e3·18-s − 2.50e4·19-s + 1.64e4·20-s + 1.09e4·21-s − 3.33e4·22-s − 1.06e5·23-s − 3.33e4·24-s + 2.28e5·25-s + 1.86e5·26-s − 1.96e4·27-s − 1.20e4·28-s + ⋯
L(s)  = 1  − 1.11·2-s − 0.577·3-s + 0.232·4-s + 1.98·5-s + 0.640·6-s − 0.448·7-s + 0.852·8-s + 0.333·9-s − 2.19·10-s + 0.602·11-s − 0.134·12-s − 1.87·13-s + 0.497·14-s − 1.14·15-s − 1.17·16-s − 1.08·17-s − 0.370·18-s − 0.838·19-s + 0.459·20-s + 0.258·21-s − 0.668·22-s − 1.81·23-s − 0.492·24-s + 2.92·25-s + 2.07·26-s − 0.192·27-s − 0.104·28-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

\(L(4)\) \(\approx\) \(0.7492192592\)
\(L(\frac12)\) \(\approx\) \(0.7492192592\)
\(L(\frac{9}{2})\) 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 \( 1 + 12.5T + 128T^{2} \)
5 \( 1 - 553.T + 7.81e4T^{2} \)
7 \( 1 + 406.T + 8.23e5T^{2} \)
11 \( 1 - 2.65e3T + 1.94e7T^{2} \)
13 \( 1 + 1.48e4T + 6.27e7T^{2} \)
17 \( 1 + 2.19e4T + 4.10e8T^{2} \)
19 \( 1 + 2.50e4T + 8.93e8T^{2} \)
23 \( 1 + 1.06e5T + 3.40e9T^{2} \)
29 \( 1 + 5.46e4T + 1.72e10T^{2} \)
31 \( 1 - 1.21e5T + 2.75e10T^{2} \)
37 \( 1 + 9.23e4T + 9.49e10T^{2} \)
41 \( 1 - 1.71e5T + 1.94e11T^{2} \)
43 \( 1 + 2.03e5T + 2.71e11T^{2} \)
47 \( 1 + 9.35e5T + 5.06e11T^{2} \)
53 \( 1 - 7.24e5T + 1.17e12T^{2} \)
59 \( 1 - 1.54e6T + 2.48e12T^{2} \)
61 \( 1 - 2.23e6T + 3.14e12T^{2} \)
67 \( 1 - 1.27e6T + 6.06e12T^{2} \)
71 \( 1 - 5.38e5T + 9.09e12T^{2} \)
73 \( 1 - 5.52e6T + 1.10e13T^{2} \)
79 \( 1 + 5.40e6T + 1.92e13T^{2} \)
83 \( 1 - 2.20e6T + 2.71e13T^{2} \)
89 \( 1 + 8.77e6T + 4.42e13T^{2} \)
97 \( 1 + 6.35e6T + 8.07e13T^{2} \)
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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

−9.969059824165109599129160026697, −9.307765619112684906049817145186, −8.335580133017286979849678909246, −6.92200732586313024239057924094, −6.42336709408512147572599610478, −5.32050083055934355126998174383, −4.40361611002787975606917025741, −2.30461878927989217319644036949, −1.81908755048386591242710298158, −0.45901333878566522542356945281, 0.45901333878566522542356945281, 1.81908755048386591242710298158, 2.30461878927989217319644036949, 4.40361611002787975606917025741, 5.32050083055934355126998174383, 6.42336709408512147572599610478, 6.92200732586313024239057924094, 8.335580133017286979849678909246, 9.307765619112684906049817145186, 9.969059824165109599129160026697

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