Properties

Label 4-8092e2-1.1-c1e2-0-3
Degree $4$
Conductor $65480464$
Sign $1$
Analytic cond. $4175.09$
Root an. cond. $8.03834$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s − 4·9-s + 6·11-s − 2·13-s + 15-s − 8·19-s + 2·21-s + 2·23-s − 8·25-s − 6·27-s − 2·29-s + 3·31-s + 6·33-s + 2·35-s + 8·37-s − 2·39-s − 11·41-s − 13·43-s − 4·45-s − 8·47-s + 3·49-s − 3·53-s + 6·55-s − 8·57-s − 61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s − 4/3·9-s + 1.80·11-s − 0.554·13-s + 0.258·15-s − 1.83·19-s + 0.436·21-s + 0.417·23-s − 8/5·25-s − 1.15·27-s − 0.371·29-s + 0.538·31-s + 1.04·33-s + 0.338·35-s + 1.31·37-s − 0.320·39-s − 1.71·41-s − 1.98·43-s − 0.596·45-s − 1.16·47-s + 3/7·49-s − 0.412·53-s + 0.809·55-s − 1.05·57-s − 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65480464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65480464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65480464\)    =    \(2^{4} \cdot 7^{2} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(4175.09\)
Root analytic conductor: \(8.03834\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 65480464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
17 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 15 T + 159 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 19 T + 253 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
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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

−7.60217189889495518950797540861, −7.52679671630293185907616371274, −6.87224664566726037575593040970, −6.45335021222017708422478611301, −6.23577899914913857989647485569, −6.21010520051678210091185781284, −5.47619513006509101751156949264, −5.32200943795411119119447140587, −4.72578622749753984191552790728, −4.51853825561178375159393747385, −3.99731217790670803112153294136, −3.80365177339396365432613331841, −3.11695610449708059222826533597, −3.03670228283412313526021562945, −2.27641201960097222341600664793, −2.10782704629434548309760781595, −1.48498691625746550509108414318, −1.36435123056020122989377568942, 0, 0, 1.36435123056020122989377568942, 1.48498691625746550509108414318, 2.10782704629434548309760781595, 2.27641201960097222341600664793, 3.03670228283412313526021562945, 3.11695610449708059222826533597, 3.80365177339396365432613331841, 3.99731217790670803112153294136, 4.51853825561178375159393747385, 4.72578622749753984191552790728, 5.32200943795411119119447140587, 5.47619513006509101751156949264, 6.21010520051678210091185781284, 6.23577899914913857989647485569, 6.45335021222017708422478611301, 6.87224664566726037575593040970, 7.52679671630293185907616371274, 7.60217189889495518950797540861

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