Properties

Label 4-384e2-1.1-c1e2-0-50
Degree $4$
Conductor $147456$
Sign $-1$
Analytic cond. $9.40192$
Root an. cond. $1.75107$
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  + 2·3-s + 9-s − 4·11-s − 4·13-s − 8·23-s − 6·25-s − 4·27-s − 8·33-s − 20·37-s − 8·39-s + 16·47-s + 2·49-s + 28·59-s − 4·61-s − 16·69-s − 24·71-s + 28·73-s − 12·75-s − 11·81-s − 12·83-s − 4·97-s − 4·99-s − 4·107-s + 12·109-s − 40·111-s − 4·117-s − 10·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.66·23-s − 6/5·25-s − 0.769·27-s − 1.39·33-s − 3.28·37-s − 1.28·39-s + 2.33·47-s + 2/7·49-s + 3.64·59-s − 0.512·61-s − 1.92·69-s − 2.84·71-s + 3.27·73-s − 1.38·75-s − 1.22·81-s − 1.31·83-s − 0.406·97-s − 0.402·99-s − 0.386·107-s + 1.14·109-s − 3.79·111-s − 0.369·117-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147456\)    =    \(2^{14} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(9.40192\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 147456,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.759776534720553396196404795081, −8.728992519223720851294830795286, −8.098681382024412731658064042949, −7.62957697295837367057177448880, −7.33738117792255002879146989083, −6.80509880453900742773724680138, −5.94298026933117547539540451748, −5.37965797861855942151102168788, −5.16542238567969836254694300670, −3.98566652809305192679300882554, −3.91058908136761919842529082491, −2.97266024838274409285785943283, −2.30676446591740616355718333113, −1.98277685475006180754048515240, 0, 1.98277685475006180754048515240, 2.30676446591740616355718333113, 2.97266024838274409285785943283, 3.91058908136761919842529082491, 3.98566652809305192679300882554, 5.16542238567969836254694300670, 5.37965797861855942151102168788, 5.94298026933117547539540451748, 6.80509880453900742773724680138, 7.33738117792255002879146989083, 7.62957697295837367057177448880, 8.098681382024412731658064042949, 8.728992519223720851294830795286, 8.759776534720553396196404795081

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