Properties

Label 2-119952-1.1-c1-0-126
Degree $2$
Conductor $119952$
Sign $-1$
Analytic cond. $957.821$
Root an. cond. $30.9486$
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  − 5-s + 3·11-s − 2·13-s + 17-s + 2·19-s − 6·23-s − 4·25-s − 3·29-s − 9·31-s + 6·37-s + 12·41-s + 12·43-s + 4·47-s − 11·53-s − 3·55-s − 59-s + 8·61-s + 2·65-s − 6·67-s + 2·73-s + 11·79-s + 3·83-s − 85-s + 10·89-s − 2·95-s − 17·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.904·11-s − 0.554·13-s + 0.242·17-s + 0.458·19-s − 1.25·23-s − 4/5·25-s − 0.557·29-s − 1.61·31-s + 0.986·37-s + 1.87·41-s + 1.82·43-s + 0.583·47-s − 1.51·53-s − 0.404·55-s − 0.130·59-s + 1.02·61-s + 0.248·65-s − 0.733·67-s + 0.234·73-s + 1.23·79-s + 0.329·83-s − 0.108·85-s + 1.05·89-s − 0.205·95-s − 1.72·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119952\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(957.821\)
Root analytic conductor: \(30.9486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{119952} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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

−13.89368221178957, −13.35775084872351, −12.68933713722147, −12.28656469428100, −11.99921313044804, −11.39494688985435, −10.88011673352874, −10.63523795082971, −9.525595893289118, −9.476452336141072, −9.244116074083971, −8.090710015764792, −7.998800213371220, −7.387320770679997, −6.983421126972066, −6.208955040574211, −5.761663427701167, −5.379895046250447, −4.435624375392545, −4.030491255048116, −3.738401335363023, −2.859209108458670, −2.260547549993113, −1.598991073433748, −0.8150811212954380, 0, 0.8150811212954380, 1.598991073433748, 2.260547549993113, 2.859209108458670, 3.738401335363023, 4.030491255048116, 4.435624375392545, 5.379895046250447, 5.761663427701167, 6.208955040574211, 6.983421126972066, 7.387320770679997, 7.998800213371220, 8.090710015764792, 9.244116074083971, 9.476452336141072, 9.525595893289118, 10.63523795082971, 10.88011673352874, 11.39494688985435, 11.99921313044804, 12.28656469428100, 12.68933713722147, 13.35775084872351, 13.89368221178957

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