Properties

Label 4-768e2-1.1-c1e2-0-23
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $37.6076$
Root an. cond. $2.47639$
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  + 4·7-s − 9-s + 12·17-s + 8·23-s + 10·25-s − 20·31-s + 4·41-s + 24·47-s − 2·49-s − 4·63-s − 8·71-s + 20·73-s + 12·79-s + 81-s − 4·89-s − 12·97-s − 20·103-s − 28·113-s + 48·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + ⋯
L(s)  = 1  + 1.51·7-s − 1/3·9-s + 2.91·17-s + 1.66·23-s + 2·25-s − 3.59·31-s + 0.624·41-s + 3.50·47-s − 2/7·49-s − 0.503·63-s − 0.949·71-s + 2.34·73-s + 1.35·79-s + 1/9·81-s − 0.423·89-s − 1.21·97-s − 1.97·103-s − 2.63·113-s + 4.40·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.970·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(37.6076\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{768} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 589824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.876822015\)
\(L(\frac12)\) \(\approx\) \(2.876822015\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−10.71952455842196254021447580282, −10.32746818568192312839748502723, −9.368856051707355902230072640025, −9.305349933171789530809623443936, −8.980194246803884730916866604967, −8.275147052231148820384204512104, −7.937663423525615724717844559466, −7.62636888582029667729632876830, −6.99709311734244714856378557170, −6.98167379803965498680802887920, −5.83696971187077453581934174752, −5.56960999106934779616024988406, −5.08102881373563885577068223802, −5.03844313933873952201198379022, −4.01758698833620399433183443014, −3.61326847361635129245552726008, −2.99890171503348173611991854582, −2.38832576738429158256579444906, −1.36953664321006418648610239834, −1.05358884779538257442246104043, 1.05358884779538257442246104043, 1.36953664321006418648610239834, 2.38832576738429158256579444906, 2.99890171503348173611991854582, 3.61326847361635129245552726008, 4.01758698833620399433183443014, 5.03844313933873952201198379022, 5.08102881373563885577068223802, 5.56960999106934779616024988406, 5.83696971187077453581934174752, 6.98167379803965498680802887920, 6.99709311734244714856378557170, 7.62636888582029667729632876830, 7.937663423525615724717844559466, 8.275147052231148820384204512104, 8.980194246803884730916866604967, 9.305349933171789530809623443936, 9.368856051707355902230072640025, 10.32746818568192312839748502723, 10.71952455842196254021447580282

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