Properties

Label 4-180e2-1.1-c0e2-0-0
Degree $4$
Conductor $32400$
Sign $1$
Analytic cond. $0.00806973$
Root an. cond. $0.299719$
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 + 7-s + 8-s + 10-s − 14-s + 15-s − 16-s − 21-s + 23-s − 24-s + 27-s + 29-s − 30-s − 35-s − 40-s + 41-s + 42-s − 2·43-s − 46-s + 47-s + 48-s + 49-s − 54-s + 56-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 14-s + 15-s − 16-s − 21-s + 23-s − 24-s + 27-s + 29-s − 30-s − 35-s − 40-s + 41-s + 42-s − 2·43-s − 46-s + 47-s + 48-s + 49-s − 54-s + 56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.00806973\)
Root analytic conductor: \(0.299719\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{180} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1855957153\)
\(L(\frac12)\) \(\approx\) \(0.1855957153\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−13.04264661200903517342027650101, −12.42240255709205703010474074280, −11.81595518153823601415363561867, −11.73600259250111435577981040744, −10.91868286944090228996622822922, −10.90274568229085661893260839840, −10.36180860361938804868043823832, −9.612274076556058093848870083957, −9.144102951616797963774964463606, −8.310140351924579553413081198973, −8.286011076617848398893161331147, −7.72987037568855243508650243723, −6.85589382162551048678061698581, −6.73967792274990793223112557608, −5.44771769571650859231311693028, −5.24204552749825469170490668341, −4.43443877732487047943431891381, −3.96139403068010698317325041094, −2.69175475848050787785423695874, −1.22605061796298853307315365905, 1.22605061796298853307315365905, 2.69175475848050787785423695874, 3.96139403068010698317325041094, 4.43443877732487047943431891381, 5.24204552749825469170490668341, 5.44771769571650859231311693028, 6.73967792274990793223112557608, 6.85589382162551048678061698581, 7.72987037568855243508650243723, 8.286011076617848398893161331147, 8.310140351924579553413081198973, 9.144102951616797963774964463606, 9.612274076556058093848870083957, 10.36180860361938804868043823832, 10.90274568229085661893260839840, 10.91868286944090228996622822922, 11.73600259250111435577981040744, 11.81595518153823601415363561867, 12.42240255709205703010474074280, 13.04264661200903517342027650101

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