Properties

Label 4-840e2-1.1-c1e2-0-35
Degree $4$
Conductor $705600$
Sign $1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s + 2·7-s − 3·8-s − 9-s + 2·10-s + 4·11-s + 4·13-s + 2·14-s − 16-s − 18-s − 2·20-s + 4·22-s − 25-s + 4·26-s − 2·28-s + 4·31-s + 5·32-s + 4·35-s + 36-s − 6·40-s − 8·43-s − 4·44-s − 2·45-s + 8·47-s − 3·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 1.06·8-s − 1/3·9-s + 0.632·10-s + 1.20·11-s + 1.10·13-s + 0.534·14-s − 1/4·16-s − 0.235·18-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.784·26-s − 0.377·28-s + 0.718·31-s + 0.883·32-s + 0.676·35-s + 1/6·36-s − 0.948·40-s − 1.21·43-s − 0.603·44-s − 0.298·45-s + 1.16·47-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.452882180\)
\(L(\frac12)\) \(\approx\) \(3.452882180\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
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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

−8.446328045598197414839717497256, −7.976539091142170111779063995556, −7.43078043385245241155123684566, −6.69602455368320265430099884426, −6.32585814320264511125955804253, −6.06691363008169893200917306256, −5.54632058713349018909600648619, −5.10233629456581752422514848965, −4.61579312894150784665818192268, −4.09085970859838519459570325008, −3.61920863905061097994083613736, −3.13669915383194461971665287674, −2.29199390294410061914888438071, −1.64566947322809716907864812723, −0.889836121605344330768338201070, 0.889836121605344330768338201070, 1.64566947322809716907864812723, 2.29199390294410061914888438071, 3.13669915383194461971665287674, 3.61920863905061097994083613736, 4.09085970859838519459570325008, 4.61579312894150784665818192268, 5.10233629456581752422514848965, 5.54632058713349018909600648619, 6.06691363008169893200917306256, 6.32585814320264511125955804253, 6.69602455368320265430099884426, 7.43078043385245241155123684566, 7.976539091142170111779063995556, 8.446328045598197414839717497256

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