Properties

Degree $4$
Conductor $21609$
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  + 2-s + 3-s + 2·4-s − 2·5-s + 6-s + 5·8-s − 2·10-s − 4·11-s + 2·12-s + 4·13-s − 2·15-s + 5·16-s − 6·17-s + 4·19-s − 4·20-s − 4·22-s + 5·24-s + 5·25-s + 4·26-s − 27-s − 4·29-s − 2·30-s + 10·32-s − 4·33-s − 6·34-s − 6·37-s + 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.408·6-s + 1.76·8-s − 0.632·10-s − 1.20·11-s + 0.577·12-s + 1.10·13-s − 0.516·15-s + 5/4·16-s − 1.45·17-s + 0.917·19-s − 0.894·20-s − 0.852·22-s + 1.02·24-s + 25-s + 0.784·26-s − 0.192·27-s − 0.742·29-s − 0.365·30-s + 1.76·32-s − 0.696·33-s − 1.02·34-s − 0.986·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08678\)
\(L(\frac12)\) \(\approx\) \(2.08678\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−13.35538967567012999742132355678, −13.02330501545202800489756566632, −12.37706777544349875402830591154, −11.75036984020560182464243259419, −11.19055537214842297399625531298, −11.03457210395954380294007920053, −10.46146611385686836717395235055, −9.950414922461296240777473019533, −8.986821541781670224412933437681, −8.525092380382120147497930696518, −7.84630499662072424196307054559, −7.66247519775145860938089500169, −6.79861725246347443155001270562, −6.56478569624224668215301513432, −5.37487154127442046284491509799, −4.98665985961532829464182856583, −4.13814467201104539154440020783, −3.53248899785197925579273264205, −2.75783616986689583278023535180, −1.77125131081290015271079329676, 1.77125131081290015271079329676, 2.75783616986689583278023535180, 3.53248899785197925579273264205, 4.13814467201104539154440020783, 4.98665985961532829464182856583, 5.37487154127442046284491509799, 6.56478569624224668215301513432, 6.79861725246347443155001270562, 7.66247519775145860938089500169, 7.84630499662072424196307054559, 8.525092380382120147497930696518, 8.986821541781670224412933437681, 9.950414922461296240777473019533, 10.46146611385686836717395235055, 11.03457210395954380294007920053, 11.19055537214842297399625531298, 11.75036984020560182464243259419, 12.37706777544349875402830591154, 13.02330501545202800489756566632, 13.35538967567012999742132355678

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