Properties

Label 4-3920e2-1.1-c0e2-0-6
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
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  − 5-s + 9-s + 4·29-s + 4·41-s − 45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s − 121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 5-s + 9-s + 4·29-s + 4·41-s − 45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s − 121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.545164227\)
\(L(\frac12)\) \(\approx\) \(1.545164227\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_1$ \( ( 1 - T )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$ \( ( 1 - T )^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.702492116265349537293042649560, −8.476584581521341309333890046944, −7.955275465023248645659869494878, −7.77915737582202487182340114251, −7.32983875188193982980792942525, −7.24802992507568953803685822014, −6.44553819269419844744212034115, −6.42174566478572999440638066040, −6.00618644976148158386440807334, −5.51556473355296007245095231125, −4.71309788299406934760416163815, −4.68396227160948575842711348784, −4.31122266114823230233121323007, −4.06327471098678505270898339461, −3.21928571733310263959036936247, −3.17364617860842158216506235416, −2.45327454368982615807571498187, −2.08894897484033772793513713653, −0.990578571973092936812892467527, −0.976050005442044870313045094085, 0.976050005442044870313045094085, 0.990578571973092936812892467527, 2.08894897484033772793513713653, 2.45327454368982615807571498187, 3.17364617860842158216506235416, 3.21928571733310263959036936247, 4.06327471098678505270898339461, 4.31122266114823230233121323007, 4.68396227160948575842711348784, 4.71309788299406934760416163815, 5.51556473355296007245095231125, 6.00618644976148158386440807334, 6.42174566478572999440638066040, 6.44553819269419844744212034115, 7.24802992507568953803685822014, 7.32983875188193982980792942525, 7.77915737582202487182340114251, 7.955275465023248645659869494878, 8.476584581521341309333890046944, 8.702492116265349537293042649560

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