Properties

Degree $4$
Conductor $36864$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 12·7-s − 5·9-s − 20·13-s − 4·19-s + 24·21-s + 18·25-s + 28·27-s − 44·31-s + 12·37-s + 40·39-s − 164·43-s + 10·49-s + 8·57-s + 172·61-s + 60·63-s − 4·67-s + 164·73-s − 36·75-s + 20·79-s − 11·81-s + 240·91-s + 88·93-s − 188·97-s − 268·103-s − 20·109-s − 24·111-s + ⋯
L(s)  = 1  − 2/3·3-s − 1.71·7-s − 5/9·9-s − 1.53·13-s − 0.210·19-s + 8/7·21-s + 0.719·25-s + 1.03·27-s − 1.41·31-s + 0.324·37-s + 1.02·39-s − 3.81·43-s + 0.204·49-s + 8/57·57-s + 2.81·61-s + 0.952·63-s − 0.0597·67-s + 2.24·73-s − 0.479·75-s + 0.253·79-s − 0.135·81-s + 2.63·91-s + 0.946·93-s − 1.93·97-s − 2.60·103-s − 0.183·109-s − 0.216·111-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Motivic weight: \(2\)
Character: induced by $\chi_{192} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.370880\)
\(L(\frac12)\) \(\approx\) \(0.370880\)
\(L(2)\) 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$C_2^2$ \( 1 - 18 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 210 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 66 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 930 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1394 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2210 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 190 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 1746 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 1554 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 86 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 5406 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 8370 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 14690 T^{2} + p^{4} T^{4} \)
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

−12.65584047590198450828423606930, −12.01950614117946585443493170285, −11.72469184887904304875096522314, −11.04149073009512687228756808992, −10.66351050982243380319133115332, −9.877753904053585087025537825215, −9.727433502065386965565122863579, −9.340386222186197727321680144535, −8.380862154553730997341495473527, −8.203195787830959945570876474767, −7.09979244414894395021587982846, −6.68819988717016081537271772800, −6.59387133494967527884977702011, −5.46015142984845819772878767787, −5.35194265588145884900693417579, −4.51940446466197251972109600854, −3.46766842693641246302661146971, −3.06375294348636723508846077756, −2.11069408651266285238567965592, −0.34949447096554236819794990158, 0.34949447096554236819794990158, 2.11069408651266285238567965592, 3.06375294348636723508846077756, 3.46766842693641246302661146971, 4.51940446466197251972109600854, 5.35194265588145884900693417579, 5.46015142984845819772878767787, 6.59387133494967527884977702011, 6.68819988717016081537271772800, 7.09979244414894395021587982846, 8.203195787830959945570876474767, 8.380862154553730997341495473527, 9.340386222186197727321680144535, 9.727433502065386965565122863579, 9.877753904053585087025537825215, 10.66351050982243380319133115332, 11.04149073009512687228756808992, 11.72469184887904304875096522314, 12.01950614117946585443493170285, 12.65584047590198450828423606930

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