Properties

Label 4-172800-1.1-c1e2-0-28
Degree $4$
Conductor $172800$
Sign $1$
Analytic cond. $11.0178$
Root an. cond. $1.82189$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 8·7-s + 9-s − 12·13-s − 8·19-s + 8·21-s + 25-s − 27-s − 12·37-s + 12·39-s + 8·43-s + 34·49-s + 8·57-s + 12·61-s − 8·63-s + 8·67-s − 28·73-s − 75-s − 32·79-s + 81-s + 96·91-s + 4·97-s − 8·103-s − 20·109-s + 12·111-s − 12·117-s − 22·121-s + ⋯
L(s)  = 1  − 0.577·3-s − 3.02·7-s + 1/3·9-s − 3.32·13-s − 1.83·19-s + 1.74·21-s + 1/5·25-s − 0.192·27-s − 1.97·37-s + 1.92·39-s + 1.21·43-s + 34/7·49-s + 1.05·57-s + 1.53·61-s − 1.00·63-s + 0.977·67-s − 3.27·73-s − 0.115·75-s − 3.60·79-s + 1/9·81-s + 10.0·91-s + 0.406·97-s − 0.788·103-s − 1.91·109-s + 1.13·111-s − 1.10·117-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(172800\)    =    \(2^{8} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(11.0178\)
Root analytic conductor: \(1.82189\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{172800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 172800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.931256392945798101264819577475, −8.312681871958482675893970014801, −7.31616880556813656693675020931, −7.18880346144848821197627995561, −6.73102949270507316916720435534, −6.46041242786302870623045801913, −5.61049576810701540702833296151, −5.45298154060946963380399619268, −4.43140184316888462520939708171, −4.19366402654330570965762069528, −3.20808194979498891977471116338, −2.75290686228331533750926083473, −2.19282779410354418054159546466, 0, 0, 2.19282779410354418054159546466, 2.75290686228331533750926083473, 3.20808194979498891977471116338, 4.19366402654330570965762069528, 4.43140184316888462520939708171, 5.45298154060946963380399619268, 5.61049576810701540702833296151, 6.46041242786302870623045801913, 6.73102949270507316916720435534, 7.18880346144848821197627995561, 7.31616880556813656693675020931, 8.312681871958482675893970014801, 8.931256392945798101264819577475

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