Properties

Label 4-192e2-1.1-c6e2-0-5
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $1951.02$
Root an. cond. $6.64608$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 484·7-s − 693·9-s − 5.23e3·13-s + 1.15e4·19-s + 2.90e3·21-s + 5.33e3·25-s + 8.53e3·27-s + 4.08e4·31-s + 9.35e4·37-s + 3.14e4·39-s + 1.37e5·43-s − 5.96e4·49-s − 6.94e4·57-s − 4.95e4·61-s + 3.35e5·63-s − 1.68e5·67-s − 2.27e5·73-s − 3.19e4·75-s + 3.19e5·79-s + 4.54e5·81-s + 2.53e6·91-s − 2.45e5·93-s + 1.79e6·97-s + 3.19e6·103-s − 4.43e6·109-s − 5.61e5·111-s + ⋯
L(s)  = 1  − 2/9·3-s − 1.41·7-s − 0.950·9-s − 2.38·13-s + 1.68·19-s + 0.313·21-s + 0.341·25-s + 0.433·27-s + 1.37·31-s + 1.84·37-s + 0.529·39-s + 1.72·43-s − 0.506·49-s − 0.374·57-s − 0.218·61-s + 1.34·63-s − 0.560·67-s − 0.585·73-s − 0.0758·75-s + 0.647·79-s + 0.854·81-s + 3.36·91-s − 0.305·93-s + 1.97·97-s + 2.92·103-s − 3.42·109-s − 0.410·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(1951.02\)
Root analytic conductor: \(6.64608\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.112126834\)
\(L(\frac12)\) \(\approx\) \(1.112126834\)
\(L(4)\) 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$C_2$ \( 1 + 2 p T + p^{6} T^{2} \)
good5$C_2^2$ \( 1 - 1066 p T^{2} + p^{12} T^{4} \)
7$C_2$ \( ( 1 + 242 T + p^{6} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 3362 p^{2} T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 + 2618 T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 1905982 T^{2} + p^{12} T^{4} \)
19$C_2$ \( ( 1 - 5786 T + p^{6} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 208876898 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 1035966962 T^{2} + p^{12} T^{4} \)
31$C_2$ \( ( 1 - 20446 T + p^{6} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 46774 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9487663202 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 - 68618 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 21106800578 T^{2} + p^{12} T^{4} \)
53$C_2^2$ \( 1 - 14819087378 T^{2} + p^{12} T^{4} \)
59$C_2^2$ \( 1 - 61991044562 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 + 24794 T + p^{6} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 84358 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 151063967522 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 + 113806 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 159742 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 388294032818 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 + 583819025758 T^{2} + p^{12} T^{4} \)
97$C_2$ \( ( 1 - 899522 T + p^{6} T^{2} )^{2} \)
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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

−11.69459031588614280336872636305, −11.44591770311912567291976852785, −10.56270965617148660508370892538, −10.14886051677067241700558332520, −9.570677948307797842425285873053, −9.453356106266141848253337074949, −8.908981435279535658268877350131, −7.918749533445129065592490459863, −7.65291288615962293471840684119, −7.11443070050069136174131212948, −6.37421571650861583044145999677, −6.08527598254909184821126892003, −5.27733038677848672352707777540, −4.88962426143364986960762175514, −4.15888718065234049936223401991, −3.01083742324964902341832456202, −2.96598959971481337731950518805, −2.29160517542408807612817766662, −0.920212159963125455458440220730, −0.36919960835207188313787989637, 0.36919960835207188313787989637, 0.920212159963125455458440220730, 2.29160517542408807612817766662, 2.96598959971481337731950518805, 3.01083742324964902341832456202, 4.15888718065234049936223401991, 4.88962426143364986960762175514, 5.27733038677848672352707777540, 6.08527598254909184821126892003, 6.37421571650861583044145999677, 7.11443070050069136174131212948, 7.65291288615962293471840684119, 7.918749533445129065592490459863, 8.908981435279535658268877350131, 9.453356106266141848253337074949, 9.570677948307797842425285873053, 10.14886051677067241700558332520, 10.56270965617148660508370892538, 11.44591770311912567291976852785, 11.69459031588614280336872636305

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