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 + 20·31-s − 22·37-s − 18·47-s + 9·49-s − 24·53-s − 6·59-s − 30·83-s + 8·103-s + 10·109-s − 36·113-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 + 3.59·31-s − 3.61·37-s − 2.62·47-s + 9/7·49-s − 3.29·53-s − 0.781·59-s − 3.29·83-s + 0.788·103-s + 0.957·109-s − 3.38·113-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

\(L(1)\) \(\approx\) \(1.245626230\)
\(L(\frac12)\) \(\approx\) \(1.245626230\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
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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

−9.200042275059527863904609124796, −8.366494203928348143144652794566, −8.323066746112530379837904426301, −7.948871936717258972109326072486, −7.24724739595439216965252383462, −6.90651622641196042568144878753, −6.56413915701137997547852606466, −6.34139643759429109096478155328, −6.21591285797117636573162204166, −5.23722392792882955840296559326, −5.16103961017549742741081044620, −4.53405262126083068144133957227, −4.47954177623568957934762654334, −3.50008211691263269420735732445, −3.21055294510375816989915517161, −2.88416416133051300556681451058, −2.74237400063408006018306758615, −1.56791815460575822623627054338, −1.27100993785628968955229239785, −0.37133089278386498690006053141, 0.37133089278386498690006053141, 1.27100993785628968955229239785, 1.56791815460575822623627054338, 2.74237400063408006018306758615, 2.88416416133051300556681451058, 3.21055294510375816989915517161, 3.50008211691263269420735732445, 4.47954177623568957934762654334, 4.53405262126083068144133957227, 5.16103961017549742741081044620, 5.23722392792882955840296559326, 6.21591285797117636573162204166, 6.34139643759429109096478155328, 6.56413915701137997547852606466, 6.90651622641196042568144878753, 7.24724739595439216965252383462, 7.948871936717258972109326072486, 8.323066746112530379837904426301, 8.366494203928348143144652794566, 9.200042275059527863904609124796

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