Properties

Label 2-471-1.1-c5-0-75
Degree $2$
Conductor $471$
Sign $-1$
Analytic cond. $75.5407$
Root an. cond. $8.69141$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.13·2-s − 9·3-s + 18.9·4-s + 54.3·5-s + 64.2·6-s − 91.4·7-s + 92.9·8-s + 81·9-s − 388.·10-s + 352.·11-s − 170.·12-s − 470.·13-s + 652.·14-s − 489.·15-s − 1.27e3·16-s + 1.82e3·17-s − 578.·18-s − 1.17e3·19-s + 1.03e3·20-s + 822.·21-s − 2.51e3·22-s − 644.·23-s − 836.·24-s − 170.·25-s + 3.36e3·26-s − 729·27-s − 1.73e3·28-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.577·3-s + 0.593·4-s + 0.972·5-s + 0.728·6-s − 0.705·7-s + 0.513·8-s + 0.333·9-s − 1.22·10-s + 0.879·11-s − 0.342·12-s − 0.772·13-s + 0.889·14-s − 0.561·15-s − 1.24·16-s + 1.52·17-s − 0.420·18-s − 0.745·19-s + 0.576·20-s + 0.407·21-s − 1.10·22-s − 0.253·23-s − 0.296·24-s − 0.0545·25-s + 0.975·26-s − 0.192·27-s − 0.418·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-1$
Analytic conductor: \(75.5407\)
Root analytic conductor: \(8.69141\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 471,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
157 \( 1 + 2.46e4T \)
good2 \( 1 + 7.13T + 32T^{2} \)
5 \( 1 - 54.3T + 3.12e3T^{2} \)
7 \( 1 + 91.4T + 1.68e4T^{2} \)
11 \( 1 - 352.T + 1.61e5T^{2} \)
13 \( 1 + 470.T + 3.71e5T^{2} \)
17 \( 1 - 1.82e3T + 1.41e6T^{2} \)
19 \( 1 + 1.17e3T + 2.47e6T^{2} \)
23 \( 1 + 644.T + 6.43e6T^{2} \)
29 \( 1 + 4.72e3T + 2.05e7T^{2} \)
31 \( 1 + 2.47e3T + 2.86e7T^{2} \)
37 \( 1 + 371.T + 6.93e7T^{2} \)
41 \( 1 - 1.62e4T + 1.15e8T^{2} \)
43 \( 1 + 1.17e4T + 1.47e8T^{2} \)
47 \( 1 - 1.31e4T + 2.29e8T^{2} \)
53 \( 1 + 1.48e4T + 4.18e8T^{2} \)
59 \( 1 - 2.85e4T + 7.14e8T^{2} \)
61 \( 1 - 2.16e3T + 8.44e8T^{2} \)
67 \( 1 - 5.57e4T + 1.35e9T^{2} \)
71 \( 1 - 1.41e4T + 1.80e9T^{2} \)
73 \( 1 + 1.03e4T + 2.07e9T^{2} \)
79 \( 1 - 8.61e4T + 3.07e9T^{2} \)
83 \( 1 + 1.90e4T + 3.93e9T^{2} \)
89 \( 1 + 6.78e3T + 5.58e9T^{2} \)
97 \( 1 + 8.79e4T + 8.58e9T^{2} \)
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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

−9.635349413004315572771980232410, −9.321624758808229767983562682996, −8.031117204202778364306829985827, −7.08061062527414401882429089551, −6.20827478839357885152275790593, −5.27890826921414352228291598123, −3.86396579462198764332330559227, −2.18240934247661418007351709561, −1.15152320757415295211033105231, 0, 1.15152320757415295211033105231, 2.18240934247661418007351709561, 3.86396579462198764332330559227, 5.27890826921414352228291598123, 6.20827478839357885152275790593, 7.08061062527414401882429089551, 8.031117204202778364306829985827, 9.321624758808229767983562682996, 9.635349413004315572771980232410

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