Properties

Label 4-442368-1.1-c1e2-0-56
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 + 4·13-s − 8·23-s + 6·25-s + 27-s − 8·33-s − 4·37-s + 4·39-s − 8·47-s − 10·49-s + 8·59-s + 28·61-s − 8·69-s − 24·71-s − 20·73-s + 6·75-s + 81-s + 24·83-s + 20·97-s − 8·99-s − 8·107-s − 12·109-s − 4·111-s + 4·117-s + 26·121-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 2.41·11-s + 1.10·13-s − 1.66·23-s + 6/5·25-s + 0.192·27-s − 1.39·33-s − 0.657·37-s + 0.640·39-s − 1.16·47-s − 1.42·49-s + 1.04·59-s + 3.58·61-s − 0.963·69-s − 2.84·71-s − 2.34·73-s + 0.692·75-s + 1/9·81-s + 2.63·83-s + 2.03·97-s − 0.804·99-s − 0.773·107-s − 1.14·109-s − 0.379·111-s + 0.369·117-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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.230854059399453444196153124887, −8.172979318844237129173419438270, −7.37312552233206901012858237940, −7.30623361332829545195635303299, −6.27798985919626463740519529757, −6.26188690510143013986588806130, −5.33300428621827304415252516413, −5.15630977188877186851978464565, −4.56007962912554840821971374527, −3.77214318657172177768570145308, −3.41295449066880137345891491973, −2.65190583901467307643981603895, −2.28349914492601444414342737921, −1.36625703357770376353073884127, 0, 1.36625703357770376353073884127, 2.28349914492601444414342737921, 2.65190583901467307643981603895, 3.41295449066880137345891491973, 3.77214318657172177768570145308, 4.56007962912554840821971374527, 5.15630977188877186851978464565, 5.33300428621827304415252516413, 6.26188690510143013986588806130, 6.27798985919626463740519529757, 7.30623361332829545195635303299, 7.37312552233206901012858237940, 8.172979318844237129173419438270, 8.230854059399453444196153124887

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