Properties

Label 4-56448-1.1-c1e2-0-0
Degree $4$
Conductor $56448$
Sign $1$
Analytic cond. $3.59917$
Root an. cond. $1.37737$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 8·19-s − 6·25-s − 4·27-s + 12·29-s + 16·31-s + 12·37-s − 7·49-s − 4·53-s + 16·57-s + 8·59-s + 12·75-s + 5·81-s − 8·83-s − 24·87-s − 32·93-s + 32·103-s − 4·109-s − 24·111-s + 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 14·147-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 1.83·19-s − 6/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s + 1.97·37-s − 49-s − 0.549·53-s + 2.11·57-s + 1.04·59-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 2.57·87-s − 3.31·93-s + 3.15·103-s − 0.383·109-s − 2.27·111-s + 3.38·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.15·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56448\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.59917\)
Root analytic conductor: \(1.37737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56448} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56448,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8788731802\)
\(L(\frac12)\) \(\approx\) \(0.8788731802\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.964671915727173343349228861429, −9.935819229981957589664665865525, −8.959311313448748386566246803110, −8.304042274266532161103040174092, −8.098990694093691505068092710868, −7.35977874808944660694472335106, −6.42897107072744719896577758454, −6.41480629060511724242965007307, −5.98894980010644060477023411018, −5.02364207902997377499660058417, −4.44992200701182918033448908949, −4.25303028692796488061253338187, −3.00889317555517420664706578512, −2.17837496002353596178446785744, −0.852162344962028721020593862686, 0.852162344962028721020593862686, 2.17837496002353596178446785744, 3.00889317555517420664706578512, 4.25303028692796488061253338187, 4.44992200701182918033448908949, 5.02364207902997377499660058417, 5.98894980010644060477023411018, 6.41480629060511724242965007307, 6.42897107072744719896577758454, 7.35977874808944660694472335106, 8.098990694093691505068092710868, 8.304042274266532161103040174092, 8.959311313448748386566246803110, 9.935819229981957589664665865525, 9.964671915727173343349228861429

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