Properties

Label 2-14490-1.1-c1-0-20
Degree $2$
Conductor $14490$
Sign $1$
Analytic cond. $115.703$
Root an. cond. $10.7565$
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 − 5-s − 7-s + 8-s − 10-s + 4·11-s + 4·13-s − 14-s + 16-s + 4·17-s + 2·19-s − 20-s + 4·22-s + 23-s + 25-s + 4·26-s − 28-s + 10·29-s + 2·31-s + 32-s + 4·34-s + 35-s + 2·37-s + 2·38-s − 40-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 1.85·29-s + 0.359·31-s + 0.176·32-s + 0.685·34-s + 0.169·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(115.703\)
Root analytic conductor: \(10.7565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14490} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 14490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.189148240\)
\(L(\frac12)\) \(\approx\) \(4.189148240\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−16.05159065010951, −15.57624611449522, −14.90854609967182, −14.36481784567580, −13.95545219394679, −13.30113527790126, −12.81260667579005, −11.99883305218104, −11.77454575570099, −11.28024783718457, −10.36510252127972, −10.01029412469557, −9.178432182384422, −8.486101442388281, −8.058987445918428, −7.064734383812872, −6.714424184824191, −6.036087740821798, −5.454616768224480, −4.537819894684605, −4.008619580570402, −3.316133940703052, −2.835297404656269, −1.503525839716617, −0.8962947115751464, 0.8962947115751464, 1.503525839716617, 2.835297404656269, 3.316133940703052, 4.008619580570402, 4.537819894684605, 5.454616768224480, 6.036087740821798, 6.714424184824191, 7.064734383812872, 8.058987445918428, 8.486101442388281, 9.178432182384422, 10.01029412469557, 10.36510252127972, 11.28024783718457, 11.77454575570099, 11.99883305218104, 12.81260667579005, 13.30113527790126, 13.95545219394679, 14.36481784567580, 14.90854609967182, 15.57624611449522, 16.05159065010951

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