Properties

Label 2-142912-1.1-c1-0-39
Degree $2$
Conductor $142912$
Sign $-1$
Analytic cond. $1141.15$
Root an. cond. $33.7810$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 3·9-s − 11-s − 6·13-s − 2·17-s − 8·19-s − 25-s − 29-s + 4·31-s + 2·35-s + 2·37-s − 2·41-s − 4·43-s − 6·45-s + 4·47-s + 49-s + 10·53-s − 2·55-s + 12·59-s − 2·61-s − 3·63-s − 12·65-s + 12·67-s − 8·71-s − 10·73-s − 77-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 9-s − 0.301·11-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 1/5·25-s − 0.185·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s + 0.583·47-s + 1/7·49-s + 1.37·53-s − 0.269·55-s + 1.56·59-s − 0.256·61-s − 0.377·63-s − 1.48·65-s + 1.46·67-s − 0.949·71-s − 1.17·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(142912\)    =    \(2^{6} \cdot 7 \cdot 11 \cdot 29\)
Sign: $-1$
Analytic conductor: \(1141.15\)
Root analytic conductor: \(33.7810\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{142912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 142912,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
29 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.49455476558962, −13.30205322239467, −12.72447019445404, −12.11761215376756, −11.79207231842395, −11.25924289607009, −10.61930086147478, −10.30487161141777, −9.840984518094288, −9.270089005943632, −8.793784972798622, −8.300903091402721, −7.915712713902716, −7.162948280562493, −6.712534778232092, −6.203836075078255, −5.541051404947979, −5.327685010357727, −4.555300386297429, −4.246236883943064, −3.332035751471061, −2.541667859493332, −2.270756545035979, −1.885911321758809, −0.7025959662799922, 0, 0.7025959662799922, 1.885911321758809, 2.270756545035979, 2.541667859493332, 3.332035751471061, 4.246236883943064, 4.555300386297429, 5.327685010357727, 5.541051404947979, 6.203836075078255, 6.712534778232092, 7.162948280562493, 7.915712713902716, 8.300903091402721, 8.793784972798622, 9.270089005943632, 9.840984518094288, 10.30487161141777, 10.61930086147478, 11.25924289607009, 11.79207231842395, 12.11761215376756, 12.72447019445404, 13.30205322239467, 13.49455476558962

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