Properties

Degree $4$
Conductor $1382976$
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  − 3-s + 2·5-s + 6·11-s + 6·13-s − 2·15-s + 4·17-s − 5·19-s + 4·23-s + 5·25-s + 27-s − 8·29-s + 7·31-s − 6·33-s + 9·37-s − 6·39-s + 4·41-s − 2·43-s + 2·47-s − 4·51-s − 8·53-s + 12·55-s + 5·57-s + 10·61-s + 12·65-s + 15·67-s − 4·69-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.80·11-s + 1.66·13-s − 0.516·15-s + 0.970·17-s − 1.14·19-s + 0.834·23-s + 25-s + 0.192·27-s − 1.48·29-s + 1.25·31-s − 1.04·33-s + 1.47·37-s − 0.960·39-s + 0.624·41-s − 0.304·43-s + 0.291·47-s − 0.560·51-s − 1.09·53-s + 1.61·55-s + 0.662·57-s + 1.28·61-s + 1.48·65-s + 1.83·67-s − 0.481·69-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1382976\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1176} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1382976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.023415167\)
\(L(\frac12)\) \(\approx\) \(3.023415167\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−9.762462760676157771304165840511, −9.731521763780025405322187399186, −9.094161503018299176577809475718, −8.897411213154396470175264085479, −8.324999790363514464437235168560, −8.203342001607343311746621593652, −7.18254928706451394834409851677, −7.14080934452002087315733315404, −6.39287676159782877920661980822, −6.21264154803786597600178023908, −5.84156156641845904224464611681, −5.61996560163064465773198465360, −4.61854071373634910023436186944, −4.57511630842210939148503411753, −3.73810090809893709000472987622, −3.50634807801491923677891080400, −2.74893910701963816799709895925, −1.99551513966424161448372955339, −1.26791861342647292081381636866, −0.951383593692832491129476640460, 0.951383593692832491129476640460, 1.26791861342647292081381636866, 1.99551513966424161448372955339, 2.74893910701963816799709895925, 3.50634807801491923677891080400, 3.73810090809893709000472987622, 4.57511630842210939148503411753, 4.61854071373634910023436186944, 5.61996560163064465773198465360, 5.84156156641845904224464611681, 6.21264154803786597600178023908, 6.39287676159782877920661980822, 7.14080934452002087315733315404, 7.18254928706451394834409851677, 8.203342001607343311746621593652, 8.324999790363514464437235168560, 8.897411213154396470175264085479, 9.094161503018299176577809475718, 9.731521763780025405322187399186, 9.762462760676157771304165840511

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