Properties

Degree $4$
Conductor $921600$
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  − 4·5-s − 9-s − 4·11-s + 16·19-s + 11·25-s + 16·29-s + 4·41-s + 4·45-s + 10·49-s + 16·55-s − 12·59-s − 4·61-s − 8·71-s + 16·79-s + 81-s − 12·89-s − 64·95-s + 4·99-s − 12·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s − 1.20·11-s + 3.67·19-s + 11/5·25-s + 2.97·29-s + 0.624·41-s + 0.596·45-s + 10/7·49-s + 2.15·55-s − 1.56·59-s − 0.512·61-s − 0.949·71-s + 1.80·79-s + 1/9·81-s − 1.27·89-s − 6.56·95-s + 0.402·99-s − 1.14·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35749\)
\(L(\frac12)\) \(\approx\) \(1.35749\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.26863020463102016845061651065, −9.856996645262923943845465558976, −9.372099576621777591253133841292, −8.948993885199513722150729850187, −8.446189733417896080430607797997, −7.971916878965730007842310926661, −7.75865510406062579704489499542, −7.50205887892589392545824838719, −6.95701144842720454553190749396, −6.61088460128492096065687493111, −5.67754109398773039329041281151, −5.52856469532631125930508084951, −4.74778390693841042544404383001, −4.70906127391465032730567318910, −3.95849213705274032262490020855, −3.14521360210025973005104356142, −3.11533291899237363460400428290, −2.61900596778946710568016469852, −1.18473864608628403330133231987, −0.64760855672245040754474559886, 0.64760855672245040754474559886, 1.18473864608628403330133231987, 2.61900596778946710568016469852, 3.11533291899237363460400428290, 3.14521360210025973005104356142, 3.95849213705274032262490020855, 4.70906127391465032730567318910, 4.74778390693841042544404383001, 5.52856469532631125930508084951, 5.67754109398773039329041281151, 6.61088460128492096065687493111, 6.95701144842720454553190749396, 7.50205887892589392545824838719, 7.75865510406062579704489499542, 7.971916878965730007842310926661, 8.446189733417896080430607797997, 8.948993885199513722150729850187, 9.372099576621777591253133841292, 9.856996645262923943845465558976, 10.26863020463102016845061651065

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