Properties

Degree $4$
Conductor $9144576$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 5·7-s − 4·11-s − 3·13-s + 7·17-s + 5·19-s − 4·23-s + 5·25-s − 29-s + 6·31-s + 10·35-s − 11·37-s − 9·41-s + 5·43-s + 6·47-s + 18·49-s + 3·53-s − 8·55-s − 14·59-s + 6·61-s − 6·65-s − 26·67-s − 16·71-s − 7·73-s − 20·77-s + 18·79-s − 83-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.88·7-s − 1.20·11-s − 0.832·13-s + 1.69·17-s + 1.14·19-s − 0.834·23-s + 25-s − 0.185·29-s + 1.07·31-s + 1.69·35-s − 1.80·37-s − 1.40·41-s + 0.762·43-s + 0.875·47-s + 18/7·49-s + 0.412·53-s − 1.07·55-s − 1.82·59-s + 0.768·61-s − 0.744·65-s − 3.17·67-s − 1.89·71-s − 0.819·73-s − 2.27·77-s + 2.02·79-s − 0.109·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.933243138\)
\(L(\frac12)\) \(\approx\) \(3.933243138\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−8.863896888237209275614346742879, −8.633031551396144018726400153143, −8.023665801355936160892343256101, −7.67263848261902962408318831021, −7.45683889328566748107510168450, −7.42200102336156823653123013723, −6.68110331953843445916008562336, −6.00339929482713966264965082087, −5.81519262853371770136578059810, −5.42001097065326163696903606582, −5.02450131863824275262345127384, −4.81478982297972312509778081544, −4.51078543401291395479326039807, −3.70704848426950133069534321467, −3.19258075270864056401353634079, −2.80539567094034626233425533875, −2.27211434018871427279312225932, −1.69032918082396636449036989377, −1.43045237854296266171254502605, −0.62795092053860399159279724238, 0.62795092053860399159279724238, 1.43045237854296266171254502605, 1.69032918082396636449036989377, 2.27211434018871427279312225932, 2.80539567094034626233425533875, 3.19258075270864056401353634079, 3.70704848426950133069534321467, 4.51078543401291395479326039807, 4.81478982297972312509778081544, 5.02450131863824275262345127384, 5.42001097065326163696903606582, 5.81519262853371770136578059810, 6.00339929482713966264965082087, 6.68110331953843445916008562336, 7.42200102336156823653123013723, 7.45683889328566748107510168450, 7.67263848261902962408318831021, 8.023665801355936160892343256101, 8.633031551396144018726400153143, 8.863896888237209275614346742879

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