Properties

Label 4-192e2-1.1-c5e2-0-2
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $948.251$
Root an. cond. $5.54920$
Motivic weight $5$
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  − 243·9-s + 2.40e3·13-s + 6.25e3·25-s − 3.31e4·37-s + 2.20e4·49-s + 7.72e4·61-s + 2.90e3·73-s + 5.90e4·81-s + 2.68e5·97-s + 2.28e5·109-s − 5.84e5·117-s − 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.59e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 9-s + 3.94·13-s + 2·25-s − 3.97·37-s + 1.31·49-s + 2.65·61-s + 0.0636·73-s + 81-s + 2.90·97-s + 1.84·109-s − 3.94·117-s − 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 9.67·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(948.251\)
Root analytic conductor: \(5.54920\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.486616555\)
\(L(\frac12)\) \(\approx\) \(3.486616555\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{5} T^{2} \)
good5$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 236 T + p^{5} T^{2} )( 1 + 236 T + p^{5} T^{2} ) \)
11$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 1202 T + p^{5} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 1432 T + p^{5} T^{2} )( 1 + 1432 T + p^{5} T^{2} ) \)
23$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10324 T + p^{5} T^{2} )( 1 + 10324 T + p^{5} T^{2} ) \)
37$C_2$ \( ( 1 + 16550 T + p^{5} T^{2} )^{2} \)
41$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 3352 T + p^{5} T^{2} )( 1 + 3352 T + p^{5} T^{2} ) \)
47$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 38626 T + p^{5} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 35536 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \)
71$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 1450 T + p^{5} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 100564 T + p^{5} T^{2} )( 1 + 100564 T + p^{5} T^{2} ) \)
83$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
89$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 134386 T + p^{5} T^{2} )^{2} \)
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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

−11.63320948851240841647611729683, −11.48423878379929271081370102813, −10.73373892615096661357399166337, −10.68415613373045265184274991948, −10.18872131841901903748712317135, −8.879290033560392649422035915079, −8.846486679487611342418607639077, −8.659706740944889884598262326530, −8.111066823814208726958604509033, −7.14856256120258757932112919024, −6.55987681479371000024761533393, −6.26996523397665687498222192207, −5.47950722604040903197899933399, −5.23191296068089890820530308938, −4.09157782044275270518137119568, −3.36950289415907175661937133932, −3.35949394079718829127442004073, −2.05652942633563333049894265028, −1.22518783736578990330557901001, −0.66476709479072142129475142476, 0.66476709479072142129475142476, 1.22518783736578990330557901001, 2.05652942633563333049894265028, 3.35949394079718829127442004073, 3.36950289415907175661937133932, 4.09157782044275270518137119568, 5.23191296068089890820530308938, 5.47950722604040903197899933399, 6.26996523397665687498222192207, 6.55987681479371000024761533393, 7.14856256120258757932112919024, 8.111066823814208726958604509033, 8.659706740944889884598262326530, 8.846486679487611342418607639077, 8.879290033560392649422035915079, 10.18872131841901903748712317135, 10.68415613373045265184274991948, 10.73373892615096661357399166337, 11.48423878379929271081370102813, 11.63320948851240841647611729683

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