Properties

Label 4-385875-1.1-c1e2-0-6
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 $1$

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: \(1\)
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

−8.560437390952492594261182815487, −8.190558210269945144592569148827, −7.980947296320422939728903060714, −7.22517929037828263493597858765, −6.40352559661425609695603895056, −5.98285749598845169856705301545, −5.47907564040983295563221596641, −5.18210179787882528820992169724, −4.25071896913475536050876516289, −3.97193527118658395993579674398, −3.61689003045514298236484607491, −3.21404721245224231470064978199, −1.96292462969238605585350779017, −1.14803979455536762352526385352, 0, 1.14803979455536762352526385352, 1.96292462969238605585350779017, 3.21404721245224231470064978199, 3.61689003045514298236484607491, 3.97193527118658395993579674398, 4.25071896913475536050876516289, 5.18210179787882528820992169724, 5.47907564040983295563221596641, 5.98285749598845169856705301545, 6.40352559661425609695603895056, 7.22517929037828263493597858765, 7.980947296320422939728903060714, 8.190558210269945144592569148827, 8.560437390952492594261182815487

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