Properties

Label 4-12e4-1.1-c1e2-0-0
Degree $4$
Conductor $20736$
Sign $1$
Analytic cond. $1.32214$
Root an. cond. $1.07230$
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  − 3·5-s − 7-s − 3·9-s + 3·11-s + 13-s + 12·17-s + 8·19-s − 3·23-s + 5·25-s − 3·29-s + 5·31-s + 3·35-s + 4·37-s − 3·41-s − 43-s + 9·45-s − 9·47-s + 7·49-s − 12·53-s − 9·55-s − 3·59-s + 13·61-s + 3·63-s − 3·65-s − 7·67-s + 24·71-s − 20·73-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s − 9-s + 0.904·11-s + 0.277·13-s + 2.91·17-s + 1.83·19-s − 0.625·23-s + 25-s − 0.557·29-s + 0.898·31-s + 0.507·35-s + 0.657·37-s − 0.468·41-s − 0.152·43-s + 1.34·45-s − 1.31·47-s + 49-s − 1.64·53-s − 1.21·55-s − 0.390·59-s + 1.66·61-s + 0.377·63-s − 0.372·65-s − 0.855·67-s + 2.84·71-s − 2.34·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20736\)    =    \(2^{8} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.32214\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9327316913\)
\(L(\frac12)\) \(\approx\) \(0.9327316913\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
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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

−13.63939936948839066671965589658, −12.49753018174528950969373641326, −12.17938940567116157236683203657, −11.98128141849072251647111542140, −11.35052065175557841697075784163, −11.17521245136674454721478943905, −10.17307402423722297079296465044, −9.686296243206800738249221035342, −9.419095640760460642674684366510, −8.451398717742178608733058238491, −8.037768318414551450244332052876, −7.71215399010766595203941265039, −7.06442974047196561827728526342, −6.27944521460285198715481266591, −5.62145106335824226285795652209, −5.13111030909990043887555761680, −4.04531457728349794313523609829, −3.34725078321806087360246920646, −3.12456873246481322824104634200, −1.10309822942172362190392132849, 1.10309822942172362190392132849, 3.12456873246481322824104634200, 3.34725078321806087360246920646, 4.04531457728349794313523609829, 5.13111030909990043887555761680, 5.62145106335824226285795652209, 6.27944521460285198715481266591, 7.06442974047196561827728526342, 7.71215399010766595203941265039, 8.037768318414551450244332052876, 8.451398717742178608733058238491, 9.419095640760460642674684366510, 9.686296243206800738249221035342, 10.17307402423722297079296465044, 11.17521245136674454721478943905, 11.35052065175557841697075784163, 11.98128141849072251647111542140, 12.17938940567116157236683203657, 12.49753018174528950969373641326, 13.63939936948839066671965589658

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