Properties

Label 4-294e2-1.1-c2e2-0-0
Degree $4$
Conductor $86436$
Sign $1$
Analytic cond. $64.1748$
Root an. cond. $2.83035$
Motivic weight $2$
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  − 2·3-s − 2·4-s − 5·9-s + 4·12-s − 2·13-s + 4·16-s − 62·19-s − 22·25-s + 28·27-s − 14·31-s + 10·36-s − 2·37-s + 4·39-s − 62·43-s − 8·48-s + 4·52-s + 124·57-s + 100·61-s − 8·64-s + 130·67-s − 194·73-s + 44·75-s + 124·76-s − 206·79-s − 11·81-s + 28·93-s − 332·97-s + ⋯
L(s)  = 1  − 2/3·3-s − 1/2·4-s − 5/9·9-s + 1/3·12-s − 0.153·13-s + 1/4·16-s − 3.26·19-s − 0.879·25-s + 1.03·27-s − 0.451·31-s + 5/18·36-s − 0.0540·37-s + 4/39·39-s − 1.44·43-s − 1/6·48-s + 1/13·52-s + 2.17·57-s + 1.63·61-s − 1/8·64-s + 1.94·67-s − 2.65·73-s + 0.586·75-s + 1.63·76-s − 2.60·79-s − 0.135·81-s + 0.301·93-s − 3.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86436\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(64.1748\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86436,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3664326912\)
\(L(\frac12)\) \(\approx\) \(0.3664326912\)
\(L(2)\) 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$C_2$ \( 1 + p T^{2} \)
3$C_2$ \( 1 + 2 T + p^{2} T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 22 T^{2} + p^{4} T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 506 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 31 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 986 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1394 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2210 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 31 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2618 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4970 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6890 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 65 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 6554 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 97 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 103 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 11978 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 1730 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 166 T + p^{2} T^{2} )^{2} \)
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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

−11.54099593810411389838942966422, −11.42246508583110982931956675307, −10.92546721175649204149241301385, −10.35917386067993420639895914539, −10.02675497105011586656715882800, −9.559721191266592519689023540103, −8.639924403211569938747714019135, −8.433890274118731173033881026709, −8.390217434238413612560123698455, −7.23967137742918327717320971194, −6.92873019094949383832012862066, −6.09308076735885301058227264457, −6.00126048529382743648643433039, −5.27471474430816613011688006242, −4.57936349551115430058826044095, −4.20952292335642111774566553367, −3.47280463764752374706074937287, −2.50449225655717400743056059183, −1.78804504103841022946141001341, −0.30063623931840240504964648974, 0.30063623931840240504964648974, 1.78804504103841022946141001341, 2.50449225655717400743056059183, 3.47280463764752374706074937287, 4.20952292335642111774566553367, 4.57936349551115430058826044095, 5.27471474430816613011688006242, 6.00126048529382743648643433039, 6.09308076735885301058227264457, 6.92873019094949383832012862066, 7.23967137742918327717320971194, 8.390217434238413612560123698455, 8.433890274118731173033881026709, 8.639924403211569938747714019135, 9.559721191266592519689023540103, 10.02675497105011586656715882800, 10.35917386067993420639895914539, 10.92546721175649204149241301385, 11.42246508583110982931956675307, 11.54099593810411389838942966422

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