Properties

Label 2-14490-1.1-c1-0-24
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 $1$

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 + 6·13-s + 14-s + 16-s − 2·17-s + 4·19-s − 20-s + 4·22-s + 23-s + 25-s − 6·26-s − 28-s + 2·29-s − 8·31-s − 32-s + 2·34-s + 35-s − 2·37-s − 4·38-s + 40-s + 6·41-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 + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-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: \(1\)
Selberg data: \((2,\ 14490,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−16.23388790478504, −15.89299104776924, −15.59354060290170, −14.88425200936766, −14.18242496980943, −13.28587914085467, −13.23369206970487, −12.40788745204283, −11.73922115293012, −11.09303623256618, −10.76535412887505, −10.18728995431144, −9.441463930797795, −8.801821405758043, −8.438785510031590, −7.610403294808575, −7.299961698759296, −6.435038372901019, −5.834683474918776, −5.199715538191292, −4.259702716954991, −3.424171544144559, −2.947532778188614, −1.927978425752178, −0.9948967426958785, 0, 0.9948967426958785, 1.927978425752178, 2.947532778188614, 3.424171544144559, 4.259702716954991, 5.199715538191292, 5.834683474918776, 6.435038372901019, 7.299961698759296, 7.610403294808575, 8.438785510031590, 8.801821405758043, 9.441463930797795, 10.18728995431144, 10.76535412887505, 11.09303623256618, 11.73922115293012, 12.40788745204283, 13.23369206970487, 13.28587914085467, 14.18242496980943, 14.88425200936766, 15.59354060290170, 15.89299104776924, 16.23388790478504

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