Properties

Label 4-3360e2-1.1-c1e2-0-4
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  − 2·3-s − 2·5-s − 2·7-s + 3·9-s − 2·11-s + 2·13-s + 4·15-s + 4·17-s − 6·19-s + 4·21-s + 3·25-s − 4·27-s − 4·29-s − 2·31-s + 4·33-s + 4·35-s − 4·37-s − 4·39-s + 4·41-s − 4·43-s − 6·45-s + 3·49-s − 8·51-s + 2·53-s + 4·55-s + 12·57-s + 8·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.970·17-s − 1.37·19-s + 0.872·21-s + 3/5·25-s − 0.769·27-s − 0.742·29-s − 0.359·31-s + 0.696·33-s + 0.676·35-s − 0.657·37-s − 0.640·39-s + 0.624·41-s − 0.609·43-s − 0.894·45-s + 3/7·49-s − 1.12·51-s + 0.274·53-s + 0.539·55-s + 1.58·57-s + 1.04·59-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

\(L(1)\) \(\approx\) \(0.9729179979\)
\(L(\frac12)\) \(\approx\) \(0.9729179979\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good11$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 2 T - 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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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

−8.685347912779028814294292898235, −8.515189343880283446517749753371, −7.895855160064608775842925002535, −7.67991509982236998473690008338, −7.28179575440451233130476195375, −6.97670119176237844922319851726, −6.38720660395664119392624550634, −6.25217297032483193589184864222, −5.89572592413273512054547858770, −5.29141503268085263519359572213, −5.14258800552124480334595810971, −4.63446697411306696495678579461, −3.98803812887003892728205493820, −3.94107491618644672873768882593, −3.26780112904927965518136479149, −3.06010293196227336532796545935, −2.08319574958456165923864023267, −1.84307111881122881287917891135, −0.73572157311878392895997360954, −0.49126058384292325145988054276, 0.49126058384292325145988054276, 0.73572157311878392895997360954, 1.84307111881122881287917891135, 2.08319574958456165923864023267, 3.06010293196227336532796545935, 3.26780112904927965518136479149, 3.94107491618644672873768882593, 3.98803812887003892728205493820, 4.63446697411306696495678579461, 5.14258800552124480334595810971, 5.29141503268085263519359572213, 5.89572592413273512054547858770, 6.25217297032483193589184864222, 6.38720660395664119392624550634, 6.97670119176237844922319851726, 7.28179575440451233130476195375, 7.67991509982236998473690008338, 7.895855160064608775842925002535, 8.515189343880283446517749753371, 8.685347912779028814294292898235

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