Properties

Label 4-18432-1.1-c1e2-0-4
Degree $4$
Conductor $18432$
Sign $1$
Analytic cond. $1.17524$
Root an. cond. $1.04119$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s + 9-s − 12·17-s − 6·25-s − 8·31-s + 4·41-s − 16·47-s + 34·49-s + 8·63-s + 32·71-s − 12·73-s − 8·79-s + 81-s + 20·89-s − 28·97-s + 24·103-s + 4·113-s − 96·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯
L(s)  = 1  + 3.02·7-s + 1/3·9-s − 2.91·17-s − 6/5·25-s − 1.43·31-s + 0.624·41-s − 2.33·47-s + 34/7·49-s + 1.00·63-s + 3.79·71-s − 1.40·73-s − 0.900·79-s + 1/9·81-s + 2.11·89-s − 2.84·97-s + 2.36·103-s + 0.376·113-s − 8.80·119-s − 0.545·121-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 + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18432\)    =    \(2^{11} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1.17524\)
Root analytic conductor: \(1.04119\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18432,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417262496\)
\(L(\frac12)\) \(\approx\) \(1.417262496\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−11.01151255843213055292894337270, −10.84849442743057013529606410604, −9.826637660192491869357018654726, −9.240123327709471504547415939693, −8.460175078145674768211752918036, −8.391046533691483698004382219288, −7.67780945046588747720337476950, −7.17212008872630953090710242182, −6.44896944953500655110243120870, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −4.62570703678849832502580328887, −3.85566973953101374011358603281, −2.09803110754033007053716721218, −1.83689405023894510739841616557, 1.83689405023894510739841616557, 2.09803110754033007053716721218, 3.85566973953101374011358603281, 4.62570703678849832502580328887, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 6.44896944953500655110243120870, 7.17212008872630953090710242182, 7.67780945046588747720337476950, 8.391046533691483698004382219288, 8.460175078145674768211752918036, 9.240123327709471504547415939693, 9.826637660192491869357018654726, 10.84849442743057013529606410604, 11.01151255843213055292894337270

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