Properties

Degree $4$
Conductor $72900$
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 + 3·5-s − 6·11-s + 16-s + 4·19-s + 3·20-s + 4·25-s + 12·29-s + 10·31-s − 12·41-s − 6·44-s − 13·49-s − 18·55-s + 24·59-s + 16·61-s + 64-s + 4·76-s + 16·79-s + 3·80-s − 36·89-s + 12·95-s + 4·100-s − 6·101-s + 4·109-s + 12·116-s + 5·121-s + 10·124-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.34·5-s − 1.80·11-s + 1/4·16-s + 0.917·19-s + 0.670·20-s + 4/5·25-s + 2.22·29-s + 1.79·31-s − 1.87·41-s − 0.904·44-s − 1.85·49-s − 2.42·55-s + 3.12·59-s + 2.04·61-s + 1/8·64-s + 0.458·76-s + 1.80·79-s + 0.335·80-s − 3.81·89-s + 1.23·95-s + 2/5·100-s − 0.597·101-s + 0.383·109-s + 1.11·116-s + 5/11·121-s + 0.898·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(72900\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{72900} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 72900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.981095907\)
\(L(\frac12)\) \(\approx\) \(1.981095907\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.872045258839871807563899217385, −9.726216919693265926792364692853, −8.592975668443455053869237962204, −8.268048909391662018895990050852, −8.018239560748665330030327379373, −7.02301072665066136063726833931, −6.73860900674537002760989556093, −6.23182947846711894237454668645, −5.37363015537505503550836847886, −5.27556992594019260057284686075, −4.64652891291450929816238213682, −3.48322127537334094984127398692, −2.62987626086243772212633915702, −2.44850978333263328730546301919, −1.21757578202308075265137076550, 1.21757578202308075265137076550, 2.44850978333263328730546301919, 2.62987626086243772212633915702, 3.48322127537334094984127398692, 4.64652891291450929816238213682, 5.27556992594019260057284686075, 5.37363015537505503550836847886, 6.23182947846711894237454668645, 6.73860900674537002760989556093, 7.02301072665066136063726833931, 8.018239560748665330030327379373, 8.268048909391662018895990050852, 8.592975668443455053869237962204, 9.726216919693265926792364692853, 9.872045258839871807563899217385

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