Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 4·19-s + 7·25-s + 12·29-s − 4·31-s + 2·37-s + 6·47-s + 9·49-s + 24·53-s − 6·59-s + 18·83-s − 16·103-s − 14·109-s + 12·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.917·19-s + 7/5·25-s + 2.22·29-s − 0.718·31-s + 0.328·37-s + 0.875·47-s + 9/7·49-s + 3.29·53-s − 0.781·59-s + 1.97·83-s − 1.57·103-s − 1.34·109-s + 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-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

\(L(1)\) \(\approx\) \(2.533109085\)
\(L(\frac12)\) \(\approx\) \(2.533109085\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 182 T^{2} + p^{2} T^{4} \)
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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

−8.843269982647401441489850758721, −8.729377528819817970608013466950, −8.165304648605752165662182017639, −7.80105336558423893799529967460, −7.22990964880727642454990844863, −7.05505944600280317167280061105, −6.58423715457298686095463840547, −6.45526467470017694434939116761, −5.86256427197644731164575718495, −5.47710033047332146544483434607, −5.17338807392832409650843043221, −4.62709529853063639464469667152, −4.09909772440257351369847562299, −3.83264971957467441980323691356, −3.06687898287322886823470819128, −3.00125135129970004035870284084, −2.54505729129247993276573365460, −1.84553571502888961989604561711, −0.865790316348362854351780989751, −0.68777287285575624213088329723, 0.68777287285575624213088329723, 0.865790316348362854351780989751, 1.84553571502888961989604561711, 2.54505729129247993276573365460, 3.00125135129970004035870284084, 3.06687898287322886823470819128, 3.83264971957467441980323691356, 4.09909772440257351369847562299, 4.62709529853063639464469667152, 5.17338807392832409650843043221, 5.47710033047332146544483434607, 5.86256427197644731164575718495, 6.45526467470017694434939116761, 6.58423715457298686095463840547, 7.05505944600280317167280061105, 7.22990964880727642454990844863, 7.80105336558423893799529967460, 8.165304648605752165662182017639, 8.729377528819817970608013466950, 8.843269982647401441489850758721

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