Properties

Degree $4$
Conductor $16384$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·5-s + 8·7-s + 6·9-s + 92·11-s − 100·13-s − 16·15-s + 92·17-s + 4·19-s + 32·21-s − 8·23-s − 46·25-s + 92·27-s − 84·29-s + 384·31-s + 368·33-s − 32·35-s + 172·37-s − 400·39-s − 300·41-s + 300·43-s − 24·45-s + 16·47-s − 446·49-s + 368·51-s + 12·53-s − 368·55-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.357·5-s + 0.431·7-s + 2/9·9-s + 2.52·11-s − 2.13·13-s − 0.275·15-s + 1.31·17-s + 0.0482·19-s + 0.332·21-s − 0.0725·23-s − 0.367·25-s + 0.655·27-s − 0.537·29-s + 2.22·31-s + 1.94·33-s − 0.154·35-s + 0.764·37-s − 1.64·39-s − 1.14·41-s + 1.06·43-s − 0.0795·45-s + 0.0496·47-s − 1.30·49-s + 1.01·51-s + 0.0311·53-s − 0.902·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16384\)    =    \(2^{14}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{128} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16384,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.08923\)
\(L(\frac12)\) \(\approx\) \(3.08923\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 62 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 8 T + 510 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 92 T + 430 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 100 T + 5166 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 92 T + 5030 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 11370 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 8798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 84 T + 45742 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 384 T + 84158 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 172 T + 99294 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 300 T + 157270 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 300 T + 180314 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 16 T + 114782 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 288382 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 644 T + 462170 T^{2} + 644 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 292 T + 240078 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 172 T + 278250 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 408 T + 672766 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 412 T + 690678 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 400 T + 963870 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 948 T + 1360138 T^{2} - 948 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 572 T + 845846 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.16154726850348948241182793717, −12.22145092644490242896657950484, −12.20685414735519986830343525635, −11.77829913500715860540719432261, −11.30207595140199057097919810845, −10.30198576382715341401256737536, −9.901445929724155115713989279880, −9.418230919452329862065684237713, −8.994498049698335360907103452765, −8.331248348067111324926838008591, −7.55580407382455872408433359252, −7.51943253153869667627807186037, −6.49360770145709687340431244092, −6.13768267618134977342485975481, −4.84312125171252364127696887047, −4.59221321474450316069971580288, −3.65871538204585852737982858969, −3.01160054344601032192191593912, −1.96205488991068877670545027130, −0.942123458980252545577404869939, 0.942123458980252545577404869939, 1.96205488991068877670545027130, 3.01160054344601032192191593912, 3.65871538204585852737982858969, 4.59221321474450316069971580288, 4.84312125171252364127696887047, 6.13768267618134977342485975481, 6.49360770145709687340431244092, 7.51943253153869667627807186037, 7.55580407382455872408433359252, 8.331248348067111324926838008591, 8.994498049698335360907103452765, 9.418230919452329862065684237713, 9.901445929724155115713989279880, 10.30198576382715341401256737536, 11.30207595140199057097919810845, 11.77829913500715860540719432261, 12.20685414735519986830343525635, 12.22145092644490242896657950484, 13.16154726850348948241182793717

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