Properties

Label 2-52983-1.1-c1-0-0
Degree $2$
Conductor $52983$
Sign $1$
Analytic cond. $423.071$
Root an. cond. $20.5686$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s + 4·11-s − 2·13-s + 14-s − 16-s − 6·17-s − 4·19-s − 2·20-s − 4·22-s − 25-s + 2·26-s + 28-s − 5·32-s + 6·34-s − 2·35-s − 6·37-s + 4·38-s + 6·40-s + 2·41-s + 4·43-s − 4·44-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.883·32-s + 1.02·34-s − 0.338·35-s − 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52983\)    =    \(3^{2} \cdot 7 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(423.071\)
Root analytic conductor: \(20.5686\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 52983,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8195184391\)
\(L(\frac12)\) \(\approx\) \(0.8195184391\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad3 \( 1 \)
7 \( 1 + T \)
29 \( 1 \)
good2 \( 1 + T + p T^{2} \) 1.2.b
5 \( 1 - 2 T + p T^{2} \) 1.5.ac
11 \( 1 - 4 T + p T^{2} \) 1.11.ae
13 \( 1 + 2 T + p T^{2} \) 1.13.c
17 \( 1 + 6 T + p T^{2} \) 1.17.g
19 \( 1 + 4 T + p T^{2} \) 1.19.e
23 \( 1 + p T^{2} \) 1.23.a
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 + 6 T + p T^{2} \) 1.37.g
41 \( 1 - 2 T + p T^{2} \) 1.41.ac
43 \( 1 - 4 T + p T^{2} \) 1.43.ae
47 \( 1 + p T^{2} \) 1.47.a
53 \( 1 + 6 T + p T^{2} \) 1.53.g
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 - 2 T + p T^{2} \) 1.61.ac
67 \( 1 - 4 T + p T^{2} \) 1.67.ae
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 - 6 T + p T^{2} \) 1.73.ag
79 \( 1 - 16 T + p T^{2} \) 1.79.aq
83 \( 1 - 12 T + p T^{2} \) 1.83.am
89 \( 1 + 14 T + p T^{2} \) 1.89.o
97 \( 1 + 18 T + p T^{2} \) 1.97.s
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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

−14.35105028040516, −13.87131600407085, −13.54081495429899, −12.99484755192615, −12.43915589679279, −12.01238849825476, −11.06682634469997, −10.82019075961379, −10.22728882024451, −9.500065155456359, −9.380310759754133, −8.941603203205257, −8.332043807860524, −7.759753555159382, −6.944186881625877, −6.579748248069606, −6.088028520538525, −5.305976908528458, −4.684854095635469, −4.146060445814317, −3.602042915769110, −2.536819246060319, −1.959174269454554, −1.376268863456180, −0.3614178923041300, 0.3614178923041300, 1.376268863456180, 1.959174269454554, 2.536819246060319, 3.602042915769110, 4.146060445814317, 4.684854095635469, 5.305976908528458, 6.088028520538525, 6.579748248069606, 6.944186881625877, 7.759753555159382, 8.332043807860524, 8.941603203205257, 9.380310759754133, 9.500065155456359, 10.22728882024451, 10.82019075961379, 11.06682634469997, 12.01238849825476, 12.43915589679279, 12.99484755192615, 13.54081495429899, 13.87131600407085, 14.35105028040516

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