Properties

Label 4-11552-1.1-c1e2-0-4
Degree $4$
Conductor $11552$
Sign $-1$
Analytic cond. $0.736565$
Root an. cond. $0.926409$
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 − 8·5-s + 8-s − 5·9-s − 8·10-s − 2·13-s + 16-s + 6·17-s − 5·18-s − 8·20-s + 38·25-s − 2·26-s − 10·29-s + 32-s + 6·34-s − 5·36-s − 4·37-s − 8·40-s − 16·41-s + 40·45-s − 5·49-s + 38·50-s − 2·52-s − 2·53-s − 10·58-s + 4·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 3.57·5-s + 0.353·8-s − 5/3·9-s − 2.52·10-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 1.17·18-s − 1.78·20-s + 38/5·25-s − 0.392·26-s − 1.85·29-s + 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.657·37-s − 1.26·40-s − 2.49·41-s + 5.96·45-s − 5/7·49-s + 5.37·50-s − 0.277·52-s − 0.274·53-s − 1.31·58-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11552\)    =    \(2^{5} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(0.736565\)
Root analytic conductor: \(0.926409\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 11552,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−11.43965443835611430533032930003, −10.98850264170077345648603277341, −10.30984639175148445762941404826, −9.250246741517628766073180669878, −8.408486909011940238981299566497, −8.199448456904276296960401284970, −7.65960330166497750263390530877, −7.25038064772684706053554512263, −6.52280948162697403172717458153, −5.28069074518443602737935578596, −5.01088818916226849614299890353, −3.86057015395193111701613139713, −3.57195691104337905967362169037, −3.01072892067414194047812488684, 0, 3.01072892067414194047812488684, 3.57195691104337905967362169037, 3.86057015395193111701613139713, 5.01088818916226849614299890353, 5.28069074518443602737935578596, 6.52280948162697403172717458153, 7.25038064772684706053554512263, 7.65960330166497750263390530877, 8.199448456904276296960401284970, 8.408486909011940238981299566497, 9.250246741517628766073180669878, 10.30984639175148445762941404826, 10.98850264170077345648603277341, 11.43965443835611430533032930003

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