Properties

Label 4-756e2-1.1-c0e2-0-1
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $0.142350$
Root an. cond. $0.614241$
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  − 7-s + 4·13-s + 19-s − 25-s + 31-s − 2·37-s − 2·43-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s − 2·97-s − 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯
L(s)  = 1  − 7-s + 4·13-s + 19-s − 25-s + 31-s − 2·37-s − 2·43-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s − 2·97-s − 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.142350\)
Root analytic conductor: \(0.614241\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{756} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9457631337\)
\(L(\frac12)\) \(\approx\) \(0.9457631337\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_1$ \( ( 1 - T )^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−10.62932871528426230517026056625, −10.50220222348165524604662537221, −9.742232412987617180579576569954, −9.687532573179586339968001977749, −8.870239416831465612952893521537, −8.699164035819916900282065218251, −8.232794775295628069526212849968, −8.034532254963809259935478754537, −7.10171719857339533757056547881, −6.78871695268480433648327619841, −6.34137702286563838020390891255, −5.96747182206304400887987792285, −5.61043948651723752723535650071, −5.04877944921169936566024468690, −4.00545553845051295599186923696, −3.92259897083775955625535401179, −3.18449586044372927020330116682, −3.10468264816957414902063507855, −1.68767077795690034784893235093, −1.26918191813805124151530000226, 1.26918191813805124151530000226, 1.68767077795690034784893235093, 3.10468264816957414902063507855, 3.18449586044372927020330116682, 3.92259897083775955625535401179, 4.00545553845051295599186923696, 5.04877944921169936566024468690, 5.61043948651723752723535650071, 5.96747182206304400887987792285, 6.34137702286563838020390891255, 6.78871695268480433648327619841, 7.10171719857339533757056547881, 8.034532254963809259935478754537, 8.232794775295628069526212849968, 8.699164035819916900282065218251, 8.870239416831465612952893521537, 9.687532573179586339968001977749, 9.742232412987617180579576569954, 10.50220222348165524604662537221, 10.62932871528426230517026056625

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