Properties

Label 4-192e2-1.1-c7e2-0-2
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $3597.35$
Root an. cond. $7.74454$
Motivic weight $7$
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  + 54·3-s − 196·5-s − 504·7-s + 2.18e3·9-s + 1.65e3·11-s − 6.15e3·13-s − 1.05e4·15-s + 1.71e4·17-s − 504·19-s − 2.72e4·21-s − 5.15e4·23-s − 2.14e4·25-s + 7.87e4·27-s + 1.99e5·29-s − 2.57e5·31-s + 8.94e4·33-s + 9.87e4·35-s + 4.68e5·37-s − 3.32e5·39-s − 1.06e5·41-s + 1.61e6·43-s − 4.28e5·45-s − 6.46e5·47-s − 5.02e5·49-s + 9.23e5·51-s − 1.46e6·53-s − 3.24e5·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.701·5-s − 0.555·7-s + 9-s + 0.375·11-s − 0.777·13-s − 0.809·15-s + 0.844·17-s − 0.0168·19-s − 0.641·21-s − 0.883·23-s − 0.274·25-s + 0.769·27-s + 1.52·29-s − 1.55·31-s + 0.433·33-s + 0.389·35-s + 1.52·37-s − 0.897·39-s − 0.242·41-s + 3.10·43-s − 0.701·45-s − 0.908·47-s − 0.610·49-s + 0.975·51-s − 1.35·53-s − 0.263·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(3597.35\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(3.648208964\)
\(L(\frac12)\) \(\approx\) \(3.648208964\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{2} \)
good5$D_{4}$ \( 1 + 196 T + 11974 p T^{2} + 196 p^{7} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 + 72 p T + 756734 T^{2} + 72 p^{8} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 - 1656 T + 35844502 T^{2} - 1656 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 6156 T + 134547182 T^{2} + 6156 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 17108 T + 486445766 T^{2} - 17108 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 + 504 T - 230552314 T^{2} + 504 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 51552 T + 7470237646 T^{2} + 51552 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 - 199804 T + 32156236718 T^{2} - 199804 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 + 257256 T + 69638832206 T^{2} + 257256 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 - 468724 T + 243553103934 T^{2} - 468724 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 + 106940 T + 285959228726 T^{2} + 106940 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 1617336 T + 1147122174038 T^{2} - 1617336 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 + 646416 T + 220226541790 T^{2} + 646416 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 1469492 T + 1664819509406 T^{2} + 1469492 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 4541544 T + 9906553362358 T^{2} - 4541544 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 - 7892 p T + 286057908078 T^{2} - 7892 p^{8} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 4775256 T + 15866190670886 T^{2} - 4775256 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 1094400 T + 3637256504686 T^{2} - 1094400 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 5731884 T + 372597068198 p T^{2} + 5731884 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 10402776 T + 57467964116462 T^{2} - 10402776 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 2212200 T + 33756154600198 T^{2} - 2212200 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 + 3604364 T + 80061377244278 T^{2} + 3604364 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 + 7156188 T + 162174325590662 T^{2} + 7156188 p^{7} T^{3} + p^{14} T^{4} \)
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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

−11.31985865091499847401543911327, −11.24083666351970386582887941960, −10.20565704701933492852205237785, −9.975360451634325881842137912065, −9.462209534053238644969582028524, −9.110164568275481413817547793101, −8.281718195853893395880278052325, −8.096394304144746429303253030070, −7.43625692238067919758258477707, −7.17563423912580281261305530235, −6.36061867721738699988513067191, −5.86449329472675767302835346336, −4.97161471212477710085636156966, −4.36943244910672222331985892154, −3.68994063153019109085578076875, −3.44572562012960974054556228570, −2.51067641928418375318586327457, −2.18505674684068188566291474748, −1.09381648684703272650126100422, −0.49639429830121355748545167606, 0.49639429830121355748545167606, 1.09381648684703272650126100422, 2.18505674684068188566291474748, 2.51067641928418375318586327457, 3.44572562012960974054556228570, 3.68994063153019109085578076875, 4.36943244910672222331985892154, 4.97161471212477710085636156966, 5.86449329472675767302835346336, 6.36061867721738699988513067191, 7.17563423912580281261305530235, 7.43625692238067919758258477707, 8.096394304144746429303253030070, 8.281718195853893395880278052325, 9.110164568275481413817547793101, 9.462209534053238644969582028524, 9.975360451634325881842137912065, 10.20565704701933492852205237785, 11.24083666351970386582887941960, 11.31985865091499847401543911327

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