↑

Properties

Label 2-135240-1.1-c1-0-23
Degree $2$
Conductor $135240$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
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  + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s − 2·17-s − 4·19-s − 23-s + 25-s + 27-s + 6·29-s + 8·31-s − 4·33-s − 2·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s − 2·51-s + 6·53-s + 4·55-s − 4·57-s + 12·59-s + 10·61-s − 2·65-s + 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 0.280·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.283488905\)
\(L(\frac12)\) \(\approx\) \(2.283488905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−13.33260328045159, −13.10244895078333, −12.59665320754950, −11.99641826111675, −11.58898162982649, −10.91456392666314, −10.58123091410791, −10.06151293618425, −9.694587493644105, −8.840649459760095, −8.453077714706294, −8.221238612526409, −7.716442118177490, −6.975376538136959, −6.653739934468152, −6.044844289647073, −5.326132647117304, −4.868986079367019, −4.131223829002178, −3.930560327141730, −3.010886647184806, −2.590516116473808, −2.123490539753824, −1.166503532173099, −0.4623992145187855, 0.4623992145187855, 1.166503532173099, 2.123490539753824, 2.590516116473808, 3.010886647184806, 3.930560327141730, 4.131223829002178, 4.868986079367019, 5.326132647117304, 6.044844289647073, 6.653739934468152, 6.975376538136959, 7.716442118177490, 8.221238612526409, 8.453077714706294, 8.840649459760095, 9.694587493644105, 10.06151293618425, 10.58123091410791, 10.91456392666314, 11.58898162982649, 11.99641826111675, 12.59665320754950, 13.10244895078333, 13.33260328045159

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