Properties

Degree $4$
Conductor $2916$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·7-s − 8·13-s + 16-s + 4·19-s − 25-s − 2·28-s + 10·31-s + 4·37-s − 20·43-s − 11·49-s − 8·52-s + 16·61-s + 64-s + 28·67-s − 14·73-s + 4·76-s + 16·79-s + 16·91-s − 2·97-s − 100-s − 8·103-s + 4·109-s − 2·112-s − 13·121-s + 10·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 1/5·25-s − 0.377·28-s + 1.79·31-s + 0.657·37-s − 3.04·43-s − 1.57·49-s − 1.10·52-s + 2.04·61-s + 1/8·64-s + 3.42·67-s − 1.63·73-s + 0.458·76-s + 1.80·79-s + 1.67·91-s − 0.203·97-s − 0.0999·100-s − 0.788·103-s + 0.383·109-s − 0.188·112-s − 1.18·121-s + 0.898·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2916} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2916,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7229881866\)
\(L(\frac12)\) \(\approx\) \(0.7229881866\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−12.98699917725805927719758339046, −11.98990591115535383596132902997, −11.91184860007480462763991644584, −11.21947207780374695340874205378, −10.11526477822370291644410044895, −9.872045258839871807563899217385, −9.512741326522107701681952133776, −8.268048909391662018895990050852, −7.82723843094092867969941089213, −6.73860900674537002760989556093, −6.71508725988629390136097161445, −5.37363015537505503550836847886, −4.77709333112371596512170551343, −3.38549178369413487864105189864, −2.44850978333263328730546301919, 2.44850978333263328730546301919, 3.38549178369413487864105189864, 4.77709333112371596512170551343, 5.37363015537505503550836847886, 6.71508725988629390136097161445, 6.73860900674537002760989556093, 7.82723843094092867969941089213, 8.268048909391662018895990050852, 9.512741326522107701681952133776, 9.872045258839871807563899217385, 10.11526477822370291644410044895, 11.21947207780374695340874205378, 11.91184860007480462763991644584, 11.98990591115535383596132902997, 12.98699917725805927719758339046

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