Properties

Label 4-384e2-1.1-c2e2-0-0
Degree $4$
Conductor $147456$
Sign $1$
Analytic cond. $109.479$
Root an. cond. $3.23469$
Motivic weight $2$
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 − 5·9-s − 28·11-s − 50·25-s + 28·27-s + 56·33-s − 98·49-s − 164·59-s + 284·73-s + 100·75-s − 11·81-s − 316·83-s − 188·97-s + 140·99-s − 356·107-s + 346·121-s + 127-s + 131-s + 137-s + 139-s + 196·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + ⋯
L(s)  = 1  − 2/3·3-s − 5/9·9-s − 2.54·11-s − 2·25-s + 1.03·27-s + 1.69·33-s − 2·49-s − 2.77·59-s + 3.89·73-s + 4/3·75-s − 0.135·81-s − 3.80·83-s − 1.93·97-s + 1.41·99-s − 3.32·107-s + 2.85·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4/3·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147456\)    =    \(2^{14} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(109.479\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 147456,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1239004201\)
\(L(\frac12)\) \(\approx\) \(0.1239004201\)
\(L(2)\) 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$C_2$ \( 1 + 2 T + p^{2} T^{2} \)
good5$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \)
19$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
31$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \)
43$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
67$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )( 1 + 62 T + p^{2} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
83$C_2$ \( ( 1 + 158 T + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 + 94 T + p^{2} T^{2} )^{2} \)
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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

−11.30259641724257649280160975024, −10.75024794439927071395623252280, −10.73426967616729772657117154096, −9.998819029082505393775656228938, −9.635821794474120245576439409597, −9.230670982289797891026353391341, −8.187346667490695667509054109090, −8.120293936511278772590669494619, −7.88266636554958245247308278228, −7.09025044442200834980042089307, −6.56317823631906136196960172286, −5.81750940026169335368597904249, −5.65684783431656092019290611163, −5.07074281665342037675154981021, −4.65851734025707899421764575577, −3.81232051520960213740770621051, −2.96466223133164404152645670631, −2.58210186700597502667321017953, −1.66206459270635103636849070426, −0.15471589191163126114192393654, 0.15471589191163126114192393654, 1.66206459270635103636849070426, 2.58210186700597502667321017953, 2.96466223133164404152645670631, 3.81232051520960213740770621051, 4.65851734025707899421764575577, 5.07074281665342037675154981021, 5.65684783431656092019290611163, 5.81750940026169335368597904249, 6.56317823631906136196960172286, 7.09025044442200834980042089307, 7.88266636554958245247308278228, 8.120293936511278772590669494619, 8.187346667490695667509054109090, 9.230670982289797891026353391341, 9.635821794474120245576439409597, 9.998819029082505393775656228938, 10.73426967616729772657117154096, 10.75024794439927071395623252280, 11.30259641724257649280160975024

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