Properties

Label 4-117e2-1.1-c1e2-0-5
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 9·7-s − 5·13-s − 6·19-s + 10·25-s − 18·28-s − 12·37-s + 13·43-s + 47·49-s + 10·52-s − 13·61-s + 8·64-s − 21·67-s + 12·76-s + 26·79-s − 45·91-s − 33·97-s − 20·100-s − 26·103-s − 11·121-s + 127-s + 131-s − 54·133-s + 137-s + 139-s + 24·148-s + 149-s + ⋯
L(s)  = 1  − 4-s + 3.40·7-s − 1.38·13-s − 1.37·19-s + 2·25-s − 3.40·28-s − 1.97·37-s + 1.98·43-s + 47/7·49-s + 1.38·52-s − 1.66·61-s + 64-s − 2.56·67-s + 1.37·76-s + 2.92·79-s − 4.71·91-s − 3.35·97-s − 2·100-s − 2.56·103-s − 121-s + 0.0887·127-s + 0.0873·131-s − 4.68·133-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-s + 0.0819·149-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: \(0\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.104410767\)
\(L(\frac12)\) \(\approx\) \(1.104410767\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−13.81279173160257393401615679087, −13.62694379516947677042840371391, −12.46427648873793078139292504378, −12.37128003121127970556062776597, −11.80718845771292300073660906113, −11.01560413161178701888136302047, −10.69394454445487923535021362642, −10.50640332215871967047634143278, −9.162552205587415278270033255433, −9.067439377155730272574142174468, −8.328349868159648652891769166710, −8.047159372711070323146310603239, −7.41249293760598380020186165924, −6.78510443580637797647023510823, −5.43648123126854160264107828925, −5.10354134532696297849656395253, −4.44536589169225904437065558856, −4.35334440313498098980548492336, −2.51377898048347983367857045794, −1.56257140765442249388862088029, 1.56257140765442249388862088029, 2.51377898048347983367857045794, 4.35334440313498098980548492336, 4.44536589169225904437065558856, 5.10354134532696297849656395253, 5.43648123126854160264107828925, 6.78510443580637797647023510823, 7.41249293760598380020186165924, 8.047159372711070323146310603239, 8.328349868159648652891769166710, 9.067439377155730272574142174468, 9.162552205587415278270033255433, 10.50640332215871967047634143278, 10.69394454445487923535021362642, 11.01560413161178701888136302047, 11.80718845771292300073660906113, 12.37128003121127970556062776597, 12.46427648873793078139292504378, 13.62694379516947677042840371391, 13.81279173160257393401615679087

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