Properties

Label 4-1176e2-1.1-c0e2-0-0
Degree $4$
Conductor $1382976$
Sign $1$
Analytic cond. $0.344452$
Root an. cond. $0.766094$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 5-s − 6-s + 8-s + 10-s + 11-s − 15-s − 16-s − 22-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 31-s + 33-s − 40-s − 48-s − 50-s + 53-s + 54-s − 55-s + 2·58-s − 59-s + 62-s + 64-s + ⋯
L(s)  = 1  − 2-s + 3-s − 5-s − 6-s + 8-s + 10-s + 11-s − 15-s − 16-s − 22-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 31-s + 33-s − 40-s − 48-s − 50-s + 53-s + 54-s − 55-s + 2·58-s − 59-s + 62-s + 64-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1382976\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.344452\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1176} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1382976,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6025527996\)
\(L(\frac12)\) \(\approx\) \(0.6025527996\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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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

−10.11649467267522829734197393932, −9.460920323510715531673400701629, −9.185071813372851386745157497706, −8.907037776560684756546142192240, −8.820130259655560051029766663259, −8.123674128320539566041894563446, −7.75414314137893028240538963046, −7.48543079372627703802208122715, −7.32380998864990173117437317453, −6.42905678795096232028728501384, −6.34137612611786791548647542959, −5.30194883798291579962600069860, −5.16674276485991631866735976195, −4.33627918375669845845352337523, −3.95038864210435977563621553114, −3.53830671774421151814635561205, −3.22029161399103696854180876123, −2.10145150397294409097414992252, −1.92269843499679796701520053419, −0.796102723187690449342926660761, 0.796102723187690449342926660761, 1.92269843499679796701520053419, 2.10145150397294409097414992252, 3.22029161399103696854180876123, 3.53830671774421151814635561205, 3.95038864210435977563621553114, 4.33627918375669845845352337523, 5.16674276485991631866735976195, 5.30194883798291579962600069860, 6.34137612611786791548647542959, 6.42905678795096232028728501384, 7.32380998864990173117437317453, 7.48543079372627703802208122715, 7.75414314137893028240538963046, 8.123674128320539566041894563446, 8.820130259655560051029766663259, 8.907037776560684756546142192240, 9.185071813372851386745157497706, 9.460920323510715531673400701629, 10.11649467267522829734197393932

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