Properties

Label 4-2e14-1.1-c3e2-0-5
Degree $4$
Conductor $16384$
Sign $1$
Analytic cond. $57.0363$
Root an. cond. $2.74813$
Motivic weight $3$
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  + 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: $1$
Analytic conductor: \(57.0363\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16384,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.922499118\)
\(L(\frac12)\) \(\approx\) \(3.922499118\)
\(L(\frac{5}{2})\) 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

−13.16151332510959660812793448131, −12.71294547861133488243939133042, −11.95218233620490209762578951165, −11.83923611797951373656279152319, −10.85748455458594214494715837847, −10.73529812629860444352909986815, −9.738122581831264796196262602920, −9.313960582537650658246706482761, −9.043305270717892583752902907929, −8.448906535760360994184158615452, −7.934139585744887966745795652895, −6.97979601724114239466939299649, −6.54501406195446523068143946475, −6.04956831398730683416393005564, −5.36196371718961953678413956835, −4.14131811639009938023043855460, −3.57225340736541773100852426548, −3.28791328557975573283912374552, −1.69347022606794249819762804062, −1.18789024419627928853200199750, 1.18789024419627928853200199750, 1.69347022606794249819762804062, 3.28791328557975573283912374552, 3.57225340736541773100852426548, 4.14131811639009938023043855460, 5.36196371718961953678413956835, 6.04956831398730683416393005564, 6.54501406195446523068143946475, 6.97979601724114239466939299649, 7.934139585744887966745795652895, 8.448906535760360994184158615452, 9.043305270717892583752902907929, 9.313960582537650658246706482761, 9.738122581831264796196262602920, 10.73529812629860444352909986815, 10.85748455458594214494715837847, 11.83923611797951373656279152319, 11.95218233620490209762578951165, 12.71294547861133488243939133042, 13.16151332510959660812793448131

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