Properties

Label 2-471-1.1-c7-0-96
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  + 16.3·2-s − 27·3-s + 137.·4-s + 361.·5-s − 440.·6-s + 685.·7-s + 158.·8-s + 729·9-s + 5.88e3·10-s − 2.07e3·11-s − 3.71e3·12-s − 1.38e3·13-s + 1.11e4·14-s − 9.75e3·15-s − 1.50e4·16-s − 1.49e4·17-s + 1.18e4·18-s + 5.09e4·19-s + 4.97e4·20-s − 1.85e4·21-s − 3.38e4·22-s + 1.64e4·23-s − 4.27e3·24-s + 5.24e4·25-s − 2.25e4·26-s − 1.96e4·27-s + 9.44e4·28-s + ⋯
L(s)  = 1  + 1.44·2-s − 0.577·3-s + 1.07·4-s + 1.29·5-s − 0.831·6-s + 0.755·7-s + 0.109·8-s + 0.333·9-s + 1.86·10-s − 0.469·11-s − 0.621·12-s − 0.174·13-s + 1.08·14-s − 0.746·15-s − 0.918·16-s − 0.736·17-s + 0.480·18-s + 1.70·19-s + 1.39·20-s − 0.436·21-s − 0.677·22-s + 0.282·23-s − 0.0631·24-s + 0.670·25-s − 0.251·26-s − 0.192·27-s + 0.812·28-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

\(L(4)\) \(\approx\) \(6.140254821\)
\(L(\frac12)\) \(\approx\) \(6.140254821\)
\(L(\frac{9}{2})\) 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 \( 1 - 16.3T + 128T^{2} \)
5 \( 1 - 361.T + 7.81e4T^{2} \)
7 \( 1 - 685.T + 8.23e5T^{2} \)
11 \( 1 + 2.07e3T + 1.94e7T^{2} \)
13 \( 1 + 1.38e3T + 6.27e7T^{2} \)
17 \( 1 + 1.49e4T + 4.10e8T^{2} \)
19 \( 1 - 5.09e4T + 8.93e8T^{2} \)
23 \( 1 - 1.64e4T + 3.40e9T^{2} \)
29 \( 1 - 2.04e5T + 1.72e10T^{2} \)
31 \( 1 - 1.80e5T + 2.75e10T^{2} \)
37 \( 1 - 3.23e5T + 9.49e10T^{2} \)
41 \( 1 - 6.62e3T + 1.94e11T^{2} \)
43 \( 1 - 3.99e5T + 2.71e11T^{2} \)
47 \( 1 + 4.52e4T + 5.06e11T^{2} \)
53 \( 1 + 1.52e6T + 1.17e12T^{2} \)
59 \( 1 - 1.39e6T + 2.48e12T^{2} \)
61 \( 1 + 1.92e6T + 3.14e12T^{2} \)
67 \( 1 + 2.93e6T + 6.06e12T^{2} \)
71 \( 1 - 3.88e6T + 9.09e12T^{2} \)
73 \( 1 - 5.59e5T + 1.10e13T^{2} \)
79 \( 1 - 3.97e6T + 1.92e13T^{2} \)
83 \( 1 - 5.08e6T + 2.71e13T^{2} \)
89 \( 1 - 4.43e6T + 4.42e13T^{2} \)
97 \( 1 - 6.92e6T + 8.07e13T^{2} \)
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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

−10.02979437716324320671453419950, −9.172353105181265695271876711796, −7.79483913205666795742177329888, −6.56971882599783591024809737704, −5.93094010871939748460415452127, −5.01954431605973569764679504408, −4.63614110953644549119115044295, −3.06054182814397123409630401979, −2.15959239260740852059470834778, −0.942750034520433726819742337424, 0.942750034520433726819742337424, 2.15959239260740852059470834778, 3.06054182814397123409630401979, 4.63614110953644549119115044295, 5.01954431605973569764679504408, 5.93094010871939748460415452127, 6.56971882599783591024809737704, 7.79483913205666795742177329888, 9.172353105181265695271876711796, 10.02979437716324320671453419950

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