Properties

Label 4-600e2-1.1-c1e2-0-39
Degree $4$
Conductor $360000$
Sign $-1$
Analytic cond. $22.9539$
Root an. cond. $2.18884$
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  + 9-s + 10·13-s − 12·17-s − 12·29-s + 4·37-s − 13·49-s − 24·53-s − 26·61-s + 4·73-s + 81-s − 14·97-s − 24·101-s − 14·109-s + 24·113-s + 10·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + 49·169-s + ⋯
L(s)  = 1  + 1/3·9-s + 2.77·13-s − 2.91·17-s − 2.22·29-s + 0.657·37-s − 1.85·49-s − 3.29·53-s − 3.32·61-s + 0.468·73-s + 1/9·81-s − 1.42·97-s − 2.38·101-s − 1.34·109-s + 2.25·113-s + 0.924·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(22.9539\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{360000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 360000,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.468231327893462435372408139803, −8.026266654629362330508166642807, −7.77221163592809854791946446876, −6.77234120616327244201676192765, −6.72617585632288603031226319311, −6.03058890823729073510214550931, −5.97534946548386281158286709106, −5.04294329753287530460269059022, −4.31968852301707049776817989287, −4.24382547629296535183762439094, −3.43647697037916365579697730592, −2.97413684260461707986207095036, −1.74800111788545442259234061107, −1.63891885984744417728938078317, 0, 1.63891885984744417728938078317, 1.74800111788545442259234061107, 2.97413684260461707986207095036, 3.43647697037916365579697730592, 4.24382547629296535183762439094, 4.31968852301707049776817989287, 5.04294329753287530460269059022, 5.97534946548386281158286709106, 6.03058890823729073510214550931, 6.72617585632288603031226319311, 6.77234120616327244201676192765, 7.77221163592809854791946446876, 8.026266654629362330508166642807, 8.468231327893462435372408139803

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