Properties

Label 2-14490-1.1-c1-0-2
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·13-s + 14-s + 16-s − 2·17-s − 4·19-s − 20-s − 23-s + 25-s − 2·26-s − 28-s + 6·29-s + 8·31-s − 32-s + 2·34-s + 35-s + 6·37-s + 4·38-s + 40-s − 2·41-s + 8·43-s + 46-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-s + 1.21·43-s + 0.147·46-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

\(L(1)\) \(\approx\) \(1.070084139\)
\(L(\frac12)\) \(\approx\) \(1.070084139\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.04203037996368, −15.65167477545501, −15.22789757296433, −14.54621247893213, −13.84188976452282, −13.28827509800869, −12.65797187550058, −12.02414768286108, −11.60823167257789, −10.79275191502703, −10.50947855157602, −9.814516456790807, −9.136233283490021, −8.565208750853133, −8.134552826229363, −7.449107651216661, −6.682639779153305, −6.291860601005413, −5.598974869711800, −4.432660467294875, −4.154625379844979, −3.013565835332805, −2.540995693133636, −1.433187074435075, −0.5296010383863949, 0.5296010383863949, 1.433187074435075, 2.540995693133636, 3.013565835332805, 4.154625379844979, 4.432660467294875, 5.598974869711800, 6.291860601005413, 6.682639779153305, 7.449107651216661, 8.134552826229363, 8.565208750853133, 9.136233283490021, 9.814516456790807, 10.50947855157602, 10.79275191502703, 11.60823167257789, 12.02414768286108, 12.65797187550058, 13.28827509800869, 13.84188976452282, 14.54621247893213, 15.22789757296433, 15.65167477545501, 16.04203037996368

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