Properties

Label 4-1216800-1.1-c1e2-0-49
Degree $4$
Conductor $1216800$
Sign $1$
Analytic cond. $77.5842$
Root an. cond. $2.96785$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 8-s + 9-s + 2·10-s − 2·13-s + 16-s − 12·17-s − 18-s − 2·20-s + 3·25-s + 2·26-s − 20·29-s − 32-s + 12·34-s + 36-s − 12·37-s + 2·40-s + 4·41-s − 2·45-s − 14·49-s − 3·50-s − 2·52-s − 12·53-s + 20·58-s + 12·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.554·13-s + 1/4·16-s − 2.91·17-s − 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.392·26-s − 3.71·29-s − 0.176·32-s + 2.05·34-s + 1/6·36-s − 1.97·37-s + 0.316·40-s + 0.624·41-s − 0.298·45-s − 2·49-s − 0.424·50-s − 0.277·52-s − 1.64·53-s + 2.62·58-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1216800\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(77.5842\)
Root analytic conductor: \(2.96785\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1216800,\ (\ :1/2, 1/2),\ 1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.48651432603210725392086041861, −7.13489552716111364035097044906, −7.05881776326155092462693764661, −6.27139115246731968799115000690, −6.13130217685388014086757536825, −5.09225276175166925177215351932, −5.02996251545709838502394177698, −4.33322549174996816277799124115, −3.75949148792159340735853491565, −3.53361530480953533803089286496, −2.61347722092406101275253466214, −2.01716871449082204952154497689, −1.63593081225043651097784490840, 0, 0, 1.63593081225043651097784490840, 2.01716871449082204952154497689, 2.61347722092406101275253466214, 3.53361530480953533803089286496, 3.75949148792159340735853491565, 4.33322549174996816277799124115, 5.02996251545709838502394177698, 5.09225276175166925177215351932, 6.13130217685388014086757536825, 6.27139115246731968799115000690, 7.05881776326155092462693764661, 7.13489552716111364035097044906, 7.48651432603210725392086041861

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