Properties

Label 4-1200e2-1.1-c1e2-0-26
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
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 + 3·9-s + 12·11-s − 12·17-s + 10·19-s + 4·27-s + 24·33-s − 2·43-s − 13·49-s − 24·51-s + 20·57-s − 12·59-s + 22·67-s + 4·73-s + 5·81-s − 12·83-s − 14·97-s + 36·99-s + 24·107-s + 24·113-s + 86·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 26·147-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 3.61·11-s − 2.91·17-s + 2.29·19-s + 0.769·27-s + 4.17·33-s − 0.304·43-s − 1.85·49-s − 3.36·51-s + 2.64·57-s − 1.56·59-s + 2.68·67-s + 0.468·73-s + 5/9·81-s − 1.31·83-s − 1.42·97-s + 3.61·99-s + 2.32·107-s + 2.25·113-s + 7.81·121-s + 0.0887·127-s − 0.352·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.14·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(91.8156\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1440000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.583848008\)
\(L(\frac12)\) \(\approx\) \(4.583848008\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−8.026266654629362330508166642807, −7.38580951662556061626475687417, −6.97251592485346598623736754257, −6.77234120616327244201676192765, −6.34671218087642508071319081770, −6.03058890823729073510214550931, −5.07979857479495636288703767744, −4.61333629110058855638846477864, −4.24382547629296535183762439094, −3.74053004282009465908034182416, −3.43647697037916365579697730592, −2.85121436671592400876592597239, −1.96284616627496219343558567397, −1.63891885984744417728938078317, −0.941088538538383902381260604609, 0.941088538538383902381260604609, 1.63891885984744417728938078317, 1.96284616627496219343558567397, 2.85121436671592400876592597239, 3.43647697037916365579697730592, 3.74053004282009465908034182416, 4.24382547629296535183762439094, 4.61333629110058855638846477864, 5.07979857479495636288703767744, 6.03058890823729073510214550931, 6.34671218087642508071319081770, 6.77234120616327244201676192765, 6.97251592485346598623736754257, 7.38580951662556061626475687417, 8.026266654629362330508166642807

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