Properties

Label 4-3360e2-1.1-c1e2-0-0
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 9-s + 4·11-s − 12·19-s − 25-s − 12·29-s − 20·31-s − 12·41-s − 2·45-s − 49-s + 8·55-s − 24·59-s + 28·61-s − 4·71-s − 16·79-s + 81-s − 12·89-s − 24·95-s − 4·99-s − 12·101-s + 28·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s + 1.20·11-s − 2.75·19-s − 1/5·25-s − 2.22·29-s − 3.59·31-s − 1.87·41-s − 0.298·45-s − 1/7·49-s + 1.07·55-s − 3.12·59-s + 3.58·61-s − 0.474·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s − 2.46·95-s − 0.402·99-s − 1.19·101-s + 2.68·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3081361429\)
\(L(\frac12)\) \(\approx\) \(0.3081361429\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.077982012850391101525928114207, −8.448031672138903000288655699172, −8.274512137559448455460535201643, −7.55628711068129622695664873114, −7.31629850307708500964429993834, −6.85000642018443407104843059337, −6.57160460943585065057136817172, −6.17629148876581429129953919448, −5.79431844707338327410518512145, −5.37476953136403275462481248555, −5.29514266980052317167154322762, −4.37898966101054919940999309713, −4.18834287360190066692406421460, −3.63499789754054518610993517307, −3.51718245214933408380273794475, −2.69092832449966498426826711707, −2.01100176092952639044898944376, −1.78303011523515690794133173816, −1.59462033990013488894225854547, −0.15157138060999760872668201614, 0.15157138060999760872668201614, 1.59462033990013488894225854547, 1.78303011523515690794133173816, 2.01100176092952639044898944376, 2.69092832449966498426826711707, 3.51718245214933408380273794475, 3.63499789754054518610993517307, 4.18834287360190066692406421460, 4.37898966101054919940999309713, 5.29514266980052317167154322762, 5.37476953136403275462481248555, 5.79431844707338327410518512145, 6.17629148876581429129953919448, 6.57160460943585065057136817172, 6.85000642018443407104843059337, 7.31629850307708500964429993834, 7.55628711068129622695664873114, 8.274512137559448455460535201643, 8.448031672138903000288655699172, 9.077982012850391101525928114207

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