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Properties

Label 4-117e2-1.1-c1e2-0-14
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 3·5-s + 9·6-s − 3·8-s + 6·9-s + 9·10-s − 6·11-s − 12·12-s − 5·13-s + 9·15-s + 3·16-s − 3·17-s − 18·18-s − 3·19-s − 12·20-s + 18·22-s − 6·23-s + 9·24-s + 25-s + 15·26-s − 9·27-s + 6·29-s − 27·30-s − 15·31-s − 6·32-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s − 1.06·8-s + 2·9-s + 2.84·10-s − 1.80·11-s − 3.46·12-s − 1.38·13-s + 2.32·15-s + 3/4·16-s − 0.727·17-s − 4.24·18-s − 0.688·19-s − 2.68·20-s + 3.83·22-s − 1.25·23-s + 1.83·24-s + 1/5·25-s + 2.94·26-s − 1.73·27-s + 1.11·29-s − 4.92·30-s − 2.69·31-s − 1.06·32-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$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 74 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 27 T + 332 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
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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

−12.98354886104095708706834147321, −12.35165226491345955772984980588, −12.03435267932437668437033047366, −11.66807984831276045420323313624, −10.84897546948938748494242978061, −10.56959718220958168531791237660, −10.27377683308262605034854737643, −9.780559707535456130152025476849, −8.923219890284389932905477106741, −8.480946020083334206735078351896, −7.72713435356032831754083300337, −7.47458363107256614223360335633, −7.03439552964069623066767208973, −6.05908241654551215166689785058, −5.29878391325248697301060944214, −4.75082974210359934621131928119, −3.79158084802677213002512374174, −2.15276262389819621973419206531, 0, 0, 2.15276262389819621973419206531, 3.79158084802677213002512374174, 4.75082974210359934621131928119, 5.29878391325248697301060944214, 6.05908241654551215166689785058, 7.03439552964069623066767208973, 7.47458363107256614223360335633, 7.72713435356032831754083300337, 8.480946020083334206735078351896, 8.923219890284389932905477106741, 9.780559707535456130152025476849, 10.27377683308262605034854737643, 10.56959718220958168531791237660, 10.84897546948938748494242978061, 11.66807984831276045420323313624, 12.03435267932437668437033047366, 12.35165226491345955772984980588, 12.98354886104095708706834147321

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