Properties

Label 4-40e4-1.1-c1e2-0-3
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  − 2·13-s − 6·17-s − 14·37-s − 16·41-s + 18·53-s − 24·61-s + 22·73-s − 9·81-s − 26·97-s − 4·101-s + 2·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 0.554·13-s − 1.45·17-s − 2.30·37-s − 2.49·41-s + 2.47·53-s − 3.07·61-s + 2.57·73-s − 81-s − 2.63·97-s − 0.398·101-s + 0.188·113-s + 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9195427721\)
\(L(\frac12)\) \(\approx\) \(0.9195427721\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.767903981935925295708492327207, −9.158047673976675898491476918859, −8.687258232474921851433377090122, −8.505857169049932365334168389233, −8.226234801848473967582035547826, −7.39847823269285495455918527513, −7.25980775975196894870560459596, −6.77304054145758788074451628387, −6.55043374149371581466503007737, −5.95840057996330046255882319759, −5.44955610623167730756439373762, −4.95607936423897530364484505212, −4.81645057521060485154422430691, −4.04574733067508784466632777220, −3.76485928125883358608548295484, −3.10195229467936784608381047703, −2.62941979123255925439395993849, −1.92297272797417177597877392857, −1.58776063976500662580433775881, −0.35694884930754046654729483871, 0.35694884930754046654729483871, 1.58776063976500662580433775881, 1.92297272797417177597877392857, 2.62941979123255925439395993849, 3.10195229467936784608381047703, 3.76485928125883358608548295484, 4.04574733067508784466632777220, 4.81645057521060485154422430691, 4.95607936423897530364484505212, 5.44955610623167730756439373762, 5.95840057996330046255882319759, 6.55043374149371581466503007737, 6.77304054145758788074451628387, 7.25980775975196894870560459596, 7.39847823269285495455918527513, 8.226234801848473967582035547826, 8.505857169049932365334168389233, 8.687258232474921851433377090122, 9.158047673976675898491476918859, 9.767903981935925295708492327207

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