Properties

Label 4-127008-1.1-c1e2-0-14
Degree $4$
Conductor $127008$
Sign $1$
Analytic cond. $8.09814$
Root an. cond. $1.68692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 12·13-s + 16-s − 4·17-s + 4·20-s + 2·25-s − 12·26-s + 4·29-s − 32-s + 4·34-s − 20·37-s − 4·40-s + 12·41-s + 49-s − 2·50-s + 12·52-s − 12·53-s − 4·58-s + 12·61-s + 64-s + 48·65-s − 4·68-s + 20·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 3.32·13-s + 1/4·16-s − 0.970·17-s + 0.894·20-s + 2/5·25-s − 2.35·26-s + 0.742·29-s − 0.176·32-s + 0.685·34-s − 3.28·37-s − 0.632·40-s + 1.87·41-s + 1/7·49-s − 0.282·50-s + 1.66·52-s − 1.64·53-s − 0.525·58-s + 1.53·61-s + 1/8·64-s + 5.95·65-s − 0.485·68-s + 2.34·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(127008\)    =    \(2^{5} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8.09814\)
Root analytic conductor: \(1.68692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{127008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 127008,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826911554\)
\(L(\frac12)\) \(\approx\) \(1.826911554\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.164662289745378790951914691450, −9.085422624242741773351689713512, −8.427817591055194605299416158333, −8.337652051623380123011624806396, −7.46998199559376075566652373057, −6.58813712157544804919343144385, −6.48878700424638503015501865218, −6.01899647743800981343957937964, −5.58110052520146908580921209772, −4.96240850233932644050318806427, −3.83863171110134241641513411613, −3.57539886636491016875562737772, −2.47676327970158091646809943718, −1.79365373540820646322667624458, −1.22168166913217334012046287130, 1.22168166913217334012046287130, 1.79365373540820646322667624458, 2.47676327970158091646809943718, 3.57539886636491016875562737772, 3.83863171110134241641513411613, 4.96240850233932644050318806427, 5.58110052520146908580921209772, 6.01899647743800981343957937964, 6.48878700424638503015501865218, 6.58813712157544804919343144385, 7.46998199559376075566652373057, 8.337652051623380123011624806396, 8.427817591055194605299416158333, 9.085422624242741773351689713512, 9.164662289745378790951914691450

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