Properties

Label 4-384e2-1.1-c1e2-0-47
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 + 4·5-s + 9-s − 8·15-s − 4·19-s − 8·23-s + 2·25-s + 4·27-s − 12·29-s − 12·43-s + 4·45-s + 16·47-s + 2·49-s − 12·53-s + 8·57-s − 20·67-s + 16·69-s − 24·71-s + 28·73-s − 4·75-s − 11·81-s + 24·87-s − 16·95-s − 4·97-s − 12·101-s − 32·115-s − 18·121-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 1/3·9-s − 2.06·15-s − 0.917·19-s − 1.66·23-s + 2/5·25-s + 0.769·27-s − 2.22·29-s − 1.82·43-s + 0.596·45-s + 2.33·47-s + 2/7·49-s − 1.64·53-s + 1.05·57-s − 2.44·67-s + 1.92·69-s − 2.84·71-s + 3.27·73-s − 0.461·75-s − 1.22·81-s + 2.57·87-s − 1.64·95-s − 0.406·97-s − 1.19·101-s − 2.98·115-s − 1.63·121-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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−9.215939702261382322141825805489, −8.759776534720553396196404795081, −7.984207807149970007419508597610, −7.62957697295837367057177448880, −6.80509880453900742773724680138, −6.39871498618546548325782647590, −5.94298026933117547539540451748, −5.59100833138318705450052051530, −5.37965797861855942151102168788, −4.43324404972048918127214240916, −3.98566652809305192679300882554, −2.97266024838274409285785943283, −1.98277685475006180754048515240, −1.73703685485230483844290438322, 0, 1.73703685485230483844290438322, 1.98277685475006180754048515240, 2.97266024838274409285785943283, 3.98566652809305192679300882554, 4.43324404972048918127214240916, 5.37965797861855942151102168788, 5.59100833138318705450052051530, 5.94298026933117547539540451748, 6.39871498618546548325782647590, 6.80509880453900742773724680138, 7.62957697295837367057177448880, 7.984207807149970007419508597610, 8.759776534720553396196404795081, 9.215939702261382322141825805489

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