Properties

Label 4-115200-1.1-c1e2-0-12
Degree $4$
Conductor $115200$
Sign $-1$
Analytic cond. $7.34525$
Root an. cond. $1.64627$
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  − 2·3-s − 2·5-s + 3·9-s − 4·13-s + 4·15-s − 25-s − 4·27-s + 16·31-s + 12·37-s + 8·39-s − 12·41-s + 8·43-s − 6·45-s − 14·49-s − 4·53-s + 8·65-s − 8·67-s + 16·71-s + 2·75-s − 16·79-s + 5·81-s − 8·83-s − 12·89-s − 32·93-s − 24·107-s − 24·111-s − 12·117-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s + 1.03·15-s − 1/5·25-s − 0.769·27-s + 2.87·31-s + 1.97·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s − 0.894·45-s − 2·49-s − 0.549·53-s + 0.992·65-s − 0.977·67-s + 1.89·71-s + 0.230·75-s − 1.80·79-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 3.31·93-s − 2.32·107-s − 2.27·111-s − 1.10·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(115200\)    =    \(2^{9} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(7.34525\)
Root analytic conductor: \(1.64627\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 115200,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.485176644293255200948827496129, −8.587862690772798837878925275978, −8.098990694093691505068092710868, −7.76762359658982396998333385003, −7.24118822048955879614406549551, −6.49464747752693623226397249889, −6.42897107072744719896577758454, −5.59404656438966764448272320215, −5.03318175754570901585796255083, −4.41965752635916350121091913407, −4.25303028692796488061253338187, −3.17819102794346399133173147888, −2.51429877325105654995284787549, −1.22863180219753375386002447637, 0, 1.22863180219753375386002447637, 2.51429877325105654995284787549, 3.17819102794346399133173147888, 4.25303028692796488061253338187, 4.41965752635916350121091913407, 5.03318175754570901585796255083, 5.59404656438966764448272320215, 6.42897107072744719896577758454, 6.49464747752693623226397249889, 7.24118822048955879614406549551, 7.76762359658982396998333385003, 8.098990694093691505068092710868, 8.587862690772798837878925275978, 9.485176644293255200948827496129

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