Properties

Label 4-60e4-1.1-c0e2-0-1
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $3.22789$
Root an. cond. $1.34038$
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  − 2·7-s − 2·13-s − 2·19-s − 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s + 4·91-s − 2·97-s − 4·103-s − 2·109-s + ⋯
L(s)  = 1  − 2·7-s − 2·13-s − 2·19-s − 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s + 4·91-s − 2·97-s − 4·103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3.22789\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1939785099\)
\(L(\frac12)\) \(\approx\) \(0.1939785099\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 + T + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−9.043086983049464747135441798559, −8.504376287316791111261920639139, −8.304910865210044839633835317920, −7.57639428319313693961799440129, −7.49528633884229337316987183669, −6.94056524709176694929115734238, −6.72155611191140098040230350033, −6.48155472651012750334871403465, −5.93662682883774945513389250973, −5.48529931616806908716620456779, −5.37459738163883289443872443183, −4.62010587423756121112540391670, −4.23370698798162955198106245853, −3.93289219689489016547639791972, −3.48759906384755615567839740278, −2.77367984686769122102100784281, −2.64698171673129953045298894380, −2.19428170232940965820214074653, −1.49527977101021543001190909688, −0.24081552773675493663994565581, 0.24081552773675493663994565581, 1.49527977101021543001190909688, 2.19428170232940965820214074653, 2.64698171673129953045298894380, 2.77367984686769122102100784281, 3.48759906384755615567839740278, 3.93289219689489016547639791972, 4.23370698798162955198106245853, 4.62010587423756121112540391670, 5.37459738163883289443872443183, 5.48529931616806908716620456779, 5.93662682883774945513389250973, 6.48155472651012750334871403465, 6.72155611191140098040230350033, 6.94056524709176694929115734238, 7.49528633884229337316987183669, 7.57639428319313693961799440129, 8.304910865210044839633835317920, 8.504376287316791111261920639139, 9.043086983049464747135441798559

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