Properties

Label 4-442368-1.1-c1e2-0-49
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 − 4·7-s + 9-s + 4·13-s − 16·19-s − 4·21-s + 6·25-s + 27-s + 12·31-s − 4·37-s + 4·39-s − 2·49-s − 16·57-s + 28·61-s − 4·63-s − 8·67-s − 20·73-s + 6·75-s − 20·79-s + 81-s − 16·91-s + 12·93-s + 20·97-s + 20·103-s − 12·109-s − 4·111-s + 4·117-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s − 3.67·19-s − 0.872·21-s + 6/5·25-s + 0.192·27-s + 2.15·31-s − 0.657·37-s + 0.640·39-s − 2/7·49-s − 2.11·57-s + 3.58·61-s − 0.503·63-s − 0.977·67-s − 2.34·73-s + 0.692·75-s − 2.25·79-s + 1/9·81-s − 1.67·91-s + 1.24·93-s + 2.03·97-s + 1.97·103-s − 1.14·109-s − 0.379·111-s + 0.369·117-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: Trivial
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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

−8.551716675388505933005243425858, −8.230854059399453444196153124887, −7.30623361332829545195635303299, −6.89547356553480728554873620104, −6.39174188210860038385829426400, −6.27798985919626463740519529757, −5.83450410447989994046324048602, −4.69490327290429029526959386590, −4.56007962912554840821971374527, −3.77214318657172177768570145308, −3.49343359434470738584054327342, −2.65190583901467307643981603895, −2.35029669656051858459889649700, −1.28088417651653847704023663777, 0, 1.28088417651653847704023663777, 2.35029669656051858459889649700, 2.65190583901467307643981603895, 3.49343359434470738584054327342, 3.77214318657172177768570145308, 4.56007962912554840821971374527, 4.69490327290429029526959386590, 5.83450410447989994046324048602, 6.27798985919626463740519529757, 6.39174188210860038385829426400, 6.89547356553480728554873620104, 7.30623361332829545195635303299, 8.230854059399453444196153124887, 8.551716675388505933005243425858

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