Properties

Label 2-471-1.1-c3-0-36
Degree $2$
Conductor $471$
Sign $1$
Analytic cond. $27.7898$
Root an. cond. $5.27161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.69·2-s − 3·3-s + 5.68·4-s + 8.40·5-s + 11.0·6-s + 21.0·7-s + 8.55·8-s + 9·9-s − 31.0·10-s + 34.6·11-s − 17.0·12-s + 82.9·13-s − 77.9·14-s − 25.2·15-s − 77.1·16-s + 123.·17-s − 33.2·18-s + 135.·19-s + 47.8·20-s − 63.1·21-s − 128.·22-s + 42.0·23-s − 25.6·24-s − 54.3·25-s − 306.·26-s − 27·27-s + 119.·28-s + ⋯
L(s)  = 1  − 1.30·2-s − 0.577·3-s + 0.711·4-s + 0.751·5-s + 0.755·6-s + 1.13·7-s + 0.377·8-s + 0.333·9-s − 0.983·10-s + 0.950·11-s − 0.410·12-s + 1.76·13-s − 1.48·14-s − 0.433·15-s − 1.20·16-s + 1.76·17-s − 0.436·18-s + 1.63·19-s + 0.534·20-s − 0.656·21-s − 1.24·22-s + 0.381·23-s − 0.218·24-s − 0.435·25-s − 2.31·26-s − 0.192·27-s + 0.808·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $1$
Analytic conductor: \(27.7898\)
Root analytic conductor: \(5.27161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.388644389\)
\(L(\frac12)\) \(\approx\) \(1.388644389\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
157 \( 1 + 157T \)
good2 \( 1 + 3.69T + 8T^{2} \)
5 \( 1 - 8.40T + 125T^{2} \)
7 \( 1 - 21.0T + 343T^{2} \)
11 \( 1 - 34.6T + 1.33e3T^{2} \)
13 \( 1 - 82.9T + 2.19e3T^{2} \)
17 \( 1 - 123.T + 4.91e3T^{2} \)
19 \( 1 - 135.T + 6.85e3T^{2} \)
23 \( 1 - 42.0T + 1.21e4T^{2} \)
29 \( 1 - 93.4T + 2.43e4T^{2} \)
31 \( 1 + 331.T + 2.97e4T^{2} \)
37 \( 1 - 307.T + 5.06e4T^{2} \)
41 \( 1 + 424.T + 6.89e4T^{2} \)
43 \( 1 + 291.T + 7.95e4T^{2} \)
47 \( 1 - 221.T + 1.03e5T^{2} \)
53 \( 1 - 15.1T + 1.48e5T^{2} \)
59 \( 1 - 218.T + 2.05e5T^{2} \)
61 \( 1 - 307.T + 2.26e5T^{2} \)
67 \( 1 - 2.90T + 3.00e5T^{2} \)
71 \( 1 + 184.T + 3.57e5T^{2} \)
73 \( 1 + 1.10e3T + 3.89e5T^{2} \)
79 \( 1 + 1.01e3T + 4.93e5T^{2} \)
83 \( 1 - 945.T + 5.71e5T^{2} \)
89 \( 1 - 787.T + 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−10.42539484756719371080540583721, −9.706794158015346057128317405418, −8.900106217329100788094552489099, −8.032020378972260802321732356539, −7.18394528741657293141702881215, −5.96510036149672420102292995925, −5.15771148203817598074073703112, −3.67373805375715635814049443556, −1.45889398971130720020487726986, −1.18901924776635684537599767053, 1.18901924776635684537599767053, 1.45889398971130720020487726986, 3.67373805375715635814049443556, 5.15771148203817598074073703112, 5.96510036149672420102292995925, 7.18394528741657293141702881215, 8.032020378972260802321732356539, 8.900106217329100788094552489099, 9.706794158015346057128317405418, 10.42539484756719371080540583721

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