Properties

Degree $4$
Conductor $51984$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 6·5-s + 9-s + 12·10-s + 4·13-s − 4·16-s − 2·17-s − 2·18-s − 12·20-s + 17·25-s − 8·26-s − 4·29-s + 8·32-s + 4·34-s + 2·36-s − 6·45-s + 11·49-s − 34·50-s + 8·52-s + 20·53-s + 8·58-s − 2·61-s − 8·64-s − 24·65-s − 4·68-s − 22·73-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2.68·5-s + 1/3·9-s + 3.79·10-s + 1.10·13-s − 16-s − 0.485·17-s − 0.471·18-s − 2.68·20-s + 17/5·25-s − 1.56·26-s − 0.742·29-s + 1.41·32-s + 0.685·34-s + 1/3·36-s − 0.894·45-s + 11/7·49-s − 4.80·50-s + 1.10·52-s + 2.74·53-s + 1.05·58-s − 0.256·61-s − 64-s − 2.97·65-s − 0.485·68-s − 2.57·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(51984\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{51984} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 51984,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.914070570189995801171942379165, −8.836427381447004052087018681826, −8.794614827642349780603070220632, −8.439333560047520544649775927937, −7.66505986486694333159066241147, −7.47848778675296589128891353641, −7.12629091377271134552327540430, −6.41173993136661837405396202601, −5.52282167690075554303727517698, −4.53894929342930739368620916750, −3.92764554574875840306496068176, −3.81591641511052084615077615666, −2.61182434413499280414159715388, −1.19093389013722883778513109873, 0, 1.19093389013722883778513109873, 2.61182434413499280414159715388, 3.81591641511052084615077615666, 3.92764554574875840306496068176, 4.53894929342930739368620916750, 5.52282167690075554303727517698, 6.41173993136661837405396202601, 7.12629091377271134552327540430, 7.47848778675296589128891353641, 7.66505986486694333159066241147, 8.439333560047520544649775927937, 8.794614827642349780603070220632, 8.836427381447004052087018681826, 9.914070570189995801171942379165

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