Properties

Label 4-180e2-1.1-c1e2-0-12
Degree $4$
Conductor $32400$
Sign $-1$
Analytic cond. $2.06585$
Root an. cond. $1.19887$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 4·13-s − 16-s − 4·17-s + 2·20-s + 3·25-s − 4·26-s + 4·29-s + 5·32-s − 4·34-s − 20·37-s + 6·40-s − 20·41-s − 14·49-s + 3·50-s + 4·52-s + 20·53-s + 4·58-s − 4·61-s + 7·64-s + 8·65-s + 4·68-s + 20·73-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.447·20-s + 3/5·25-s − 0.784·26-s + 0.742·29-s + 0.883·32-s − 0.685·34-s − 3.28·37-s + 0.948·40-s − 3.12·41-s − 2·49-s + 0.424·50-s + 0.554·52-s + 2.74·53-s + 0.525·58-s − 0.512·61-s + 7/8·64-s + 0.992·65-s + 0.485·68-s + 2.34·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(2.06585\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 32400,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.26700477708582759405639390127, −9.738851148469479131038939478590, −8.982199030019329801304860528684, −8.554588868470699894187777077269, −8.270140448885357284476919824336, −7.40264879408133961073539513214, −6.76955885158912340897756950945, −6.55157891837643531715592264808, −5.32469243442401148699352691061, −5.04860090093668311608273372623, −4.54153907278640853760279816056, −3.63412818236731287070311550143, −3.29035819756263414840801407793, −2.11935007719153584295308930473, 0, 2.11935007719153584295308930473, 3.29035819756263414840801407793, 3.63412818236731287070311550143, 4.54153907278640853760279816056, 5.04860090093668311608273372623, 5.32469243442401148699352691061, 6.55157891837643531715592264808, 6.76955885158912340897756950945, 7.40264879408133961073539513214, 8.270140448885357284476919824336, 8.554588868470699894187777077269, 8.982199030019329801304860528684, 9.738851148469479131038939478590, 10.26700477708582759405639390127

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