Properties

Label 4-442368-1.1-c1e2-0-69
Degree $4$
Conductor $442368$
Sign $-1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
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 + 9-s + 8·11-s − 12·13-s + 8·23-s − 10·25-s + 27-s + 8·33-s − 4·37-s − 12·39-s − 24·47-s − 10·49-s + 8·59-s − 4·61-s + 8·69-s − 8·71-s − 20·73-s − 10·75-s + 81-s − 24·83-s − 12·97-s + 8·99-s + 24·107-s + 4·109-s − 4·111-s − 12·117-s + 26·121-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 2.41·11-s − 3.32·13-s + 1.66·23-s − 2·25-s + 0.192·27-s + 1.39·33-s − 0.657·37-s − 1.92·39-s − 3.50·47-s − 1.42·49-s + 1.04·59-s − 0.512·61-s + 0.963·69-s − 0.949·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 2.63·83-s − 1.21·97-s + 0.804·99-s + 2.32·107-s + 0.383·109-s − 0.379·111-s − 1.10·117-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{442368} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−8.503024042291174545390595385787, −7.67554775307177371631314255080, −7.56237844869816331895572212869, −6.86617713649335228477722027286, −6.80910210319960057293306406499, −6.19174829158795733764599049329, −5.36858848597467135489376840463, −4.99207876302239290655432150674, −4.36074490911347519116057298154, −4.14300745369301184814326433789, −3.06807634409715093203408940942, −3.04605515966584879794132221618, −1.89290593869789230240007927288, −1.59108885184574423909298130831, 0, 1.59108885184574423909298130831, 1.89290593869789230240007927288, 3.04605515966584879794132221618, 3.06807634409715093203408940942, 4.14300745369301184814326433789, 4.36074490911347519116057298154, 4.99207876302239290655432150674, 5.36858848597467135489376840463, 6.19174829158795733764599049329, 6.80910210319960057293306406499, 6.86617713649335228477722027286, 7.56237844869816331895572212869, 7.67554775307177371631314255080, 8.503024042291174545390595385787

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