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Properties

Label 4-2925e2-1.1-c0e2-0-0
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $2.13091$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 16-s − 2·19-s + 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s − 2·109-s − 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·7-s − 16-s − 2·19-s + 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s − 2·109-s − 2·112-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.13091\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.578109028\)
\(L(\frac12)\) \(\approx\) \(1.578109028\)
\(L(1)\) 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)$
bad3 \( 1 \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−8.961648947526123855276460415419, −8.578463707715024107279476466777, −8.321117685088286588720852072272, −8.096598058912461166318883789972, −7.924785992434291661239084190691, −7.05267160860198830843268447143, −6.94351350092379117722643286356, −6.56959945734339281681644154544, −6.18498535589391991572459290462, −5.49558565105280841295918485178, −5.24270426378048592321858880200, −4.85374351795357824825049766707, −4.37794120111384212584934433152, −4.24394926466181307307903669687, −3.72720249732470050320271435260, −2.96332388599962322690076830622, −2.45039883942702676141526581431, −1.87717345643110840097775915967, −1.78382365759962944775249701589, −0.797753375799117961488007663994, 0.797753375799117961488007663994, 1.78382365759962944775249701589, 1.87717345643110840097775915967, 2.45039883942702676141526581431, 2.96332388599962322690076830622, 3.72720249732470050320271435260, 4.24394926466181307307903669687, 4.37794120111384212584934433152, 4.85374351795357824825049766707, 5.24270426378048592321858880200, 5.49558565105280841295918485178, 6.18498535589391991572459290462, 6.56959945734339281681644154544, 6.94351350092379117722643286356, 7.05267160860198830843268447143, 7.924785992434291661239084190691, 8.096598058912461166318883789972, 8.321117685088286588720852072272, 8.578463707715024107279476466777, 8.961648947526123855276460415419

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