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Properties

Label 4-117e2-1.1-c1e2-0-19
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $0.872822$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s − 6·5-s − 6·7-s + 6·9-s + 6·11-s + 6·12-s − 5·13-s + 18·15-s − 6·17-s + 12·20-s + 18·21-s − 3·23-s + 19·25-s − 9·27-s + 12·28-s − 6·29-s + 6·31-s − 18·33-s + 36·35-s − 12·36-s + 15·39-s − 12·41-s + 43-s − 12·44-s − 36·45-s − 12·47-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 2.68·5-s − 2.26·7-s + 2·9-s + 1.80·11-s + 1.73·12-s − 1.38·13-s + 4.64·15-s − 1.45·17-s + 2.68·20-s + 3.92·21-s − 0.625·23-s + 19/5·25-s − 1.73·27-s + 2.26·28-s − 1.11·29-s + 1.07·31-s − 3.13·33-s + 6.08·35-s − 2·36-s + 2.40·39-s − 1.87·41-s + 0.152·43-s − 1.80·44-s − 5.36·45-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.872822\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 131 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−12.90540293630586733890153355394, −12.64976601342064648747269327247, −12.12900601149315251790140746565, −11.75438835735683116415305978526, −11.48412798336320981215192535822, −10.94226950350821712296634159637, −10.00178610684291064617627635800, −9.681905834105698325070932089493, −9.144995044936558140044674438003, −8.477662955428252514888537980504, −7.63110419811954353490859704665, −6.99488438903327558810410830610, −6.54728851093017236777411448367, −6.28951309323056334665448752408, −4.76331302955890611710624149097, −4.64371638059730855232114849506, −3.80247328858458021016633500459, −3.47613086007436455330254655724, 0, 0, 3.47613086007436455330254655724, 3.80247328858458021016633500459, 4.64371638059730855232114849506, 4.76331302955890611710624149097, 6.28951309323056334665448752408, 6.54728851093017236777411448367, 6.99488438903327558810410830610, 7.63110419811954353490859704665, 8.477662955428252514888537980504, 9.144995044936558140044674438003, 9.681905834105698325070932089493, 10.00178610684291064617627635800, 10.94226950350821712296634159637, 11.48412798336320981215192535822, 11.75438835735683116415305978526, 12.12900601149315251790140746565, 12.64976601342064648747269327247, 12.90540293630586733890153355394

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