Properties

Label 4-1216800-1.1-c1e2-0-47
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 $1$

Origins

Origins of factors

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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 + 12·29-s + 32-s − 12·34-s + 36-s − 20·37-s + 2·40-s − 12·41-s + 2·45-s − 14·49-s + 3·50-s + 2·52-s − 20·53-s + 12·58-s − 4·61-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 + 2.22·29-s + 0.176·32-s − 2.05·34-s + 1/6·36-s − 3.28·37-s + 0.316·40-s − 1.87·41-s + 0.298·45-s − 2·49-s + 0.424·50-s + 0.277·52-s − 2.74·53-s + 1.57·58-s − 0.512·61-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: $\chi_{1216800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−7.83417942869426066771961040753, −7.03582983327921838179279968038, −6.57950396679982686721830292017, −6.57267675826487089075525795107, −6.33947776642796638344079621345, −5.43160779320744251383504417132, −5.06664273077784135902280870302, −4.53496762516360346181339328954, −4.50539658406620147914202325372, −3.42325323576864435489079625967, −3.26654328534844814185904778325, −2.46147122738666541505186218774, −1.80910327227356048364459418629, −1.55977141690761434296396620452, 0, 1.55977141690761434296396620452, 1.80910327227356048364459418629, 2.46147122738666541505186218774, 3.26654328534844814185904778325, 3.42325323576864435489079625967, 4.50539658406620147914202325372, 4.53496762516360346181339328954, 5.06664273077784135902280870302, 5.43160779320744251383504417132, 6.33947776642796638344079621345, 6.57267675826487089075525795107, 6.57950396679982686721830292017, 7.03582983327921838179279968038, 7.83417942869426066771961040753

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