Properties

Label 4-117e2-1.1-c3e2-0-0
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $47.6544$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s − 34·5-s − 20·7-s + 32·8-s − 136·10-s − 32·11-s − 91·13-s − 80·14-s + 128·16-s − 13·17-s − 30·19-s − 272·20-s − 128·22-s + 78·23-s + 617·25-s − 364·26-s − 160·28-s + 197·29-s − 148·31-s + 256·32-s − 52·34-s + 680·35-s + 227·37-s − 120·38-s − 1.08e3·40-s − 165·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 3.04·5-s − 1.07·7-s + 1.41·8-s − 4.30·10-s − 0.877·11-s − 1.94·13-s − 1.52·14-s + 2·16-s − 0.185·17-s − 0.362·19-s − 3.04·20-s − 1.24·22-s + 0.707·23-s + 4.93·25-s − 2.74·26-s − 1.07·28-s + 1.26·29-s − 0.857·31-s + 1.41·32-s − 0.262·34-s + 3.28·35-s + 1.00·37-s − 0.512·38-s − 4.30·40-s − 0.628·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(47.6544\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9011774189\)
\(L(\frac12)\) \(\approx\) \(0.9011774189\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + 7 p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \)
5$C_2$ \( ( 1 + 17 T + p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 17 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \)
11$C_2^2$ \( 1 + 32 T - 307 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 13 T - 4744 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 30 T - 5959 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 197 T + 14420 T^{2} - 197 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 74 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 227 T + 876 T^{2} - 227 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 + 165 T - 41696 T^{2} + 165 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 156 T - 55171 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2$ \( ( 1 - 162 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 + 93 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 864 T + 541117 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 862 T + 442281 T^{2} + 862 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 654 T + 69805 T^{2} - 654 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 215 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 76 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + 628 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 266 T - 634213 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 238 T - 856029 T^{2} + 238 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.87993555298852486369814418026, −12.66093998048758542024825354070, −12.46348795018452505709853834650, −11.95865268144257655709726531203, −11.53914734188059523126776617729, −10.69381267638092977288407556632, −10.66234047378761658897119268464, −9.750184455754149035134559704579, −8.925876822004409352708914737327, −7.923300015195436940336247614283, −7.87373730988957627041575501399, −7.08713352480391029442969380925, −7.04250547757241899155703179321, −5.78739518556314141562941616824, −4.74297279782872041196559999615, −4.63486439017950348776856827851, −4.00052274732900274624091171472, −3.25799616798662732723071481507, −2.72758950928515278561499549060, −0.39515562018637130489725262223, 0.39515562018637130489725262223, 2.72758950928515278561499549060, 3.25799616798662732723071481507, 4.00052274732900274624091171472, 4.63486439017950348776856827851, 4.74297279782872041196559999615, 5.78739518556314141562941616824, 7.04250547757241899155703179321, 7.08713352480391029442969380925, 7.87373730988957627041575501399, 7.923300015195436940336247614283, 8.925876822004409352708914737327, 9.750184455754149035134559704579, 10.66234047378761658897119268464, 10.69381267638092977288407556632, 11.53914734188059523126776617729, 11.95865268144257655709726531203, 12.46348795018452505709853834650, 12.66093998048758542024825354070, 13.87993555298852486369814418026

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