Properties

Label 4-10368-1.1-c1e2-0-1
Degree $4$
Conductor $10368$
Sign $1$
Analytic cond. $0.661072$
Root an. cond. $0.901700$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 4·13-s − 4·17-s + 2·25-s − 12·29-s + 12·37-s + 12·41-s − 14·49-s + 4·53-s − 4·61-s − 16·65-s + 20·73-s − 16·85-s + 12·89-s + 4·97-s + 36·101-s − 4·109-s − 36·113-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.10·13-s − 0.970·17-s + 2/5·25-s − 2.22·29-s + 1.97·37-s + 1.87·41-s − 2·49-s + 0.549·53-s − 0.512·61-s − 1.98·65-s + 2.34·73-s − 1.73·85-s + 1.27·89-s + 0.406·97-s + 3.58·101-s − 0.383·109-s − 3.38·113-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10368\)    =    \(2^{7} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.661072\)
Root analytic conductor: \(0.901700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.211782339\)
\(L(\frac12)\) \(\approx\) \(1.211782339\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−11.17432181718975973992131540327, −11.14455695554061766980709369025, −10.16546244400809085298891139382, −9.747086677582411849557090474382, −9.298977106556460794427243537385, −9.065170023749915638598404079220, −7.77541900804615864403738320707, −7.62403207119791885504595168937, −6.50812189570177474416634496690, −6.14969218484336320739052245813, −5.48498394296229820428473072147, −4.83326855247570334999188527992, −3.89741909491206350829507060388, −2.51659494204614894722398853037, −1.97363471421369880359461961552, 1.97363471421369880359461961552, 2.51659494204614894722398853037, 3.89741909491206350829507060388, 4.83326855247570334999188527992, 5.48498394296229820428473072147, 6.14969218484336320739052245813, 6.50812189570177474416634496690, 7.62403207119791885504595168937, 7.77541900804615864403738320707, 9.065170023749915638598404079220, 9.298977106556460794427243537385, 9.747086677582411849557090474382, 10.16546244400809085298891139382, 11.14455695554061766980709369025, 11.17432181718975973992131540327

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