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Properties

Label 2-471-471.470-c0-0-4
Degree $2$
Conductor $471$
Sign $1$
Analytic cond. $0.235059$
Root an. cond. $0.484829$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 3-s + 1.00·4-s + 1.41·5-s − 1.41·6-s + 9-s − 2.00·10-s + 1.00·12-s − 2·13-s + 1.41·15-s − 0.999·16-s − 1.41·18-s + 1.41·20-s − 1.41·23-s + 1.00·25-s + 2.82·26-s + 27-s − 1.41·29-s − 2.00·30-s + 1.41·32-s + 1.00·36-s − 2·39-s + 1.41·41-s + 1.41·45-s + 2.00·46-s − 0.999·48-s + ⋯
L(s)  = 1  − 1.41·2-s + 3-s + 1.00·4-s + 1.41·5-s − 1.41·6-s + 9-s − 2.00·10-s + 1.00·12-s − 2·13-s + 1.41·15-s − 0.999·16-s − 1.41·18-s + 1.41·20-s − 1.41·23-s + 1.00·25-s + 2.82·26-s + 27-s − 1.41·29-s − 2.00·30-s + 1.41·32-s + 1.00·36-s − 2·39-s + 1.41·41-s + 1.41·45-s + 2.00·46-s − 0.999·48-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $1$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (470, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7080687134\)
\(L(\frac12)\) \(\approx\) \(0.7080687134\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
157 \( 1 - T \)
good2 \( 1 + 1.41T + T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 2T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.59641431402496486636748397607, −9.917403828107444022517627536216, −9.514449135073434739269912421918, −8.840270017306984339050149154643, −7.68388746721648997216331809607, −7.18920670895652441122089601320, −5.81962419504973149652330199978, −4.44355126011627230227179175517, −2.52394046340656355362992692667, −1.84774859200377399647644055471, 1.84774859200377399647644055471, 2.52394046340656355362992692667, 4.44355126011627230227179175517, 5.81962419504973149652330199978, 7.18920670895652441122089601320, 7.68388746721648997216331809607, 8.840270017306984339050149154643, 9.514449135073434739269912421918, 9.917403828107444022517627536216, 10.59641431402496486636748397607

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