Properties

Label 4-378e2-1.1-c1e2-0-17
Degree $4$
Conductor $142884$
Sign $1$
Analytic cond. $9.11040$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·7-s + 10·13-s + 16-s + 4·19-s − 10·25-s + 2·28-s + 10·31-s + 4·37-s − 2·43-s + 3·49-s + 10·52-s − 20·61-s + 64-s − 26·67-s + 4·73-s + 4·76-s − 20·79-s + 20·91-s + 16·97-s − 10·100-s − 26·103-s − 32·109-s + 2·112-s − 22·121-s + 10·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s + 2.77·13-s + 1/4·16-s + 0.917·19-s − 2·25-s + 0.377·28-s + 1.79·31-s + 0.657·37-s − 0.304·43-s + 3/7·49-s + 1.38·52-s − 2.56·61-s + 1/8·64-s − 3.17·67-s + 0.468·73-s + 0.458·76-s − 2.25·79-s + 2.09·91-s + 1.62·97-s − 100-s − 2.56·103-s − 3.06·109-s + 0.188·112-s − 2·121-s + 0.898·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(142884\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9.11040\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{142884} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 142884,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.350635247\)
\(L(\frac12)\) \(\approx\) \(2.350635247\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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

−9.358886306281919204370388916695, −8.682143010158560978147219777821, −8.360658565008429029479731162281, −7.77295226765590417072124632150, −7.66081859530557802153309309893, −6.72994634489814968752118634163, −6.29670817383847541122613475379, −5.74408802320171077841105153966, −5.61879984028395623126782300752, −4.35441080626390222166206464206, −4.30645127647783484429164317499, −3.34773629764867655679280376645, −2.89963569250000254763681159752, −1.68869994047015348511995782895, −1.25841763926025585348683837537, 1.25841763926025585348683837537, 1.68869994047015348511995782895, 2.89963569250000254763681159752, 3.34773629764867655679280376645, 4.30645127647783484429164317499, 4.35441080626390222166206464206, 5.61879984028395623126782300752, 5.74408802320171077841105153966, 6.29670817383847541122613475379, 6.72994634489814968752118634163, 7.66081859530557802153309309893, 7.77295226765590417072124632150, 8.360658565008429029479731162281, 8.682143010158560978147219777821, 9.358886306281919204370388916695

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