Properties

Label 4-117e2-1.1-c1e2-0-1
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $0.872822$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 9-s + 2·12-s + 13-s + 12·17-s + 25-s − 27-s + 6·29-s − 2·36-s − 39-s + 4·43-s − 49-s − 12·51-s − 2·52-s + 6·53-s − 19·61-s + 8·64-s − 24·68-s − 75-s + 2·79-s + 81-s − 6·87-s − 2·100-s − 12·101-s − 23·103-s − 6·107-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s + 0.277·13-s + 2.91·17-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1/3·36-s − 0.160·39-s + 0.609·43-s − 1/7·49-s − 1.68·51-s − 0.277·52-s + 0.824·53-s − 2.43·61-s + 64-s − 2.91·68-s − 0.115·75-s + 0.225·79-s + 1/9·81-s − 0.643·87-s − 1/5·100-s − 1.19·101-s − 2.26·103-s − 0.580·107-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7504322481\)
\(L(\frac12)\) \(\approx\) \(0.7504322481\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
13$C_2$ \( 1 - T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 109 T^{2} + p^{2} T^{4} \)
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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

−11.17310265426394578567858862158, −10.62140071039119189009830844367, −10.01018128430228393562239035514, −9.734525388066678762361827150079, −9.055899324665407824822280875918, −8.444232221747720494186058647024, −7.83514408784472815960806099309, −7.34571791133079122239411951389, −6.45553358777330586185470515225, −5.77256502001201223181385340597, −5.26575184583197541866305505205, −4.59800742644128741164440299805, −3.81249724247691758317758302026, −2.98095841830027523443701648612, −1.15142841883494888918585582967, 1.15142841883494888918585582967, 2.98095841830027523443701648612, 3.81249724247691758317758302026, 4.59800742644128741164440299805, 5.26575184583197541866305505205, 5.77256502001201223181385340597, 6.45553358777330586185470515225, 7.34571791133079122239411951389, 7.83514408784472815960806099309, 8.444232221747720494186058647024, 9.055899324665407824822280875918, 9.734525388066678762361827150079, 10.01018128430228393562239035514, 10.62140071039119189009830844367, 11.17310265426394578567858862158

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