Properties

Label 4-385875-1.1-c1e2-0-8
Degree $4$
Conductor $385875$
Sign $1$
Analytic cond. $24.6037$
Root an. cond. $2.22715$
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 − 4·4-s − 5-s − 7-s − 2·9-s + 4·12-s − 10·13-s + 15-s + 12·16-s + 4·20-s + 21-s + 12·23-s + 25-s + 5·27-s + 4·28-s + 35-s + 8·36-s + 10·39-s − 24·41-s + 2·45-s − 12·48-s + 49-s + 40·52-s − 24·53-s − 4·60-s + 2·63-s − 32·64-s + ⋯
L(s)  = 1  − 0.577·3-s − 2·4-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.15·12-s − 2.77·13-s + 0.258·15-s + 3·16-s + 0.894·20-s + 0.218·21-s + 2.50·23-s + 1/5·25-s + 0.962·27-s + 0.755·28-s + 0.169·35-s + 4/3·36-s + 1.60·39-s − 3.74·41-s + 0.298·45-s − 1.73·48-s + 1/7·49-s + 5.54·52-s − 3.29·53-s − 0.516·60-s + 0.251·63-s − 4·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(385875\)    =    \(3^{2} \cdot 5^{3} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(24.6037\)
Root analytic conductor: \(2.22715\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{385875} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 385875,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_2$ \( 1 + T + p T^{2} \)
5$C_1$ \( 1 + T \)
7$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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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

−8.214592059880155258547933248387, −7.959921722640743022302973334932, −7.21456919347212808652226059968, −6.93454011637102289378074371468, −6.36641447795918613892295316962, −5.45020178597402395728815947142, −5.19374718417958536543626274426, −4.78686073935120916368506592009, −4.71225653823478357730435111839, −3.78858238344580500375502888439, −2.98704088538274633622207409794, −2.94585098531436589990293317610, −1.37785913473779166712390600334, 0, 0, 1.37785913473779166712390600334, 2.94585098531436589990293317610, 2.98704088538274633622207409794, 3.78858238344580500375502888439, 4.71225653823478357730435111839, 4.78686073935120916368506592009, 5.19374718417958536543626274426, 5.45020178597402395728815947142, 6.36641447795918613892295316962, 6.93454011637102289378074371468, 7.21456919347212808652226059968, 7.959921722640743022302973334932, 8.214592059880155258547933248387

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