Properties

Label 2-471-471.470-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(1.750968461\)
\(L(\frac12)\) \(\approx\) \(1.750968461\)
\(L(1)\) 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

−11.76922379516970855247727833107, −10.49596351853003685411057681396, −9.355874575086333827023369389758, −8.374487734677172249264361163465, −7.39273492070135982784061930222, −6.80694458765800976428616028132, −4.98841129051684845294834601243, −4.47656870152900480534318143030, −3.39217486400474163878557711673, −2.62234272501928377429498525157, 2.62234272501928377429498525157, 3.39217486400474163878557711673, 4.47656870152900480534318143030, 4.98841129051684845294834601243, 6.80694458765800976428616028132, 7.39273492070135982784061930222, 8.374487734677172249264361163465, 9.355874575086333827023369389758, 10.49596351853003685411057681396, 11.76922379516970855247727833107

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