Properties

Label 4-19008-1.1-c1e2-0-1
Degree $4$
Conductor $19008$
Sign $1$
Analytic cond. $1.21196$
Root an. cond. $1.04923$
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  − 3-s + 9-s + 11-s + 4·13-s + 2·25-s − 27-s − 33-s + 4·37-s − 4·39-s + 12·47-s + 2·49-s + 12·59-s − 8·61-s − 12·71-s − 8·73-s − 2·75-s + 81-s − 20·97-s + 99-s + 4·109-s − 4·111-s + 4·117-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 2/5·25-s − 0.192·27-s − 0.174·33-s + 0.657·37-s − 0.640·39-s + 1.75·47-s + 2/7·49-s + 1.56·59-s − 1.02·61-s − 1.42·71-s − 0.936·73-s − 0.230·75-s + 1/9·81-s − 2.03·97-s + 0.100·99-s + 0.383·109-s − 0.379·111-s + 0.369·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19008\)    =    \(2^{6} \cdot 3^{3} \cdot 11\)
Sign: $1$
Analytic conductor: \(1.21196\)
Root analytic conductor: \(1.04923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19008} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19008,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.008812861\)
\(L(\frac12)\) \(\approx\) \(1.008812861\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−10.87111583454372921305705022754, −10.57755026186379514837066820890, −9.872822078123312712017635366132, −9.339855126610289039276263015751, −8.670410637699767160646012915426, −8.342849935134971891357488978271, −7.38133321672302546420095787319, −7.07489979355103692195249240932, −6.13113280286443542004063358489, −5.94054070785065555014940896467, −5.09827818531269205265502168789, −4.29213337932568858922592669887, −3.70936690680474055080283033333, −2.63227716153411648714898260305, −1.25821432380895100299434540193, 1.25821432380895100299434540193, 2.63227716153411648714898260305, 3.70936690680474055080283033333, 4.29213337932568858922592669887, 5.09827818531269205265502168789, 5.94054070785065555014940896467, 6.13113280286443542004063358489, 7.07489979355103692195249240932, 7.38133321672302546420095787319, 8.342849935134971891357488978271, 8.670410637699767160646012915426, 9.339855126610289039276263015751, 9.872822078123312712017635366132, 10.57755026186379514837066820890, 10.87111583454372921305705022754

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