Properties

Label 4-1536e2-1.1-c1e2-0-17
Degree $4$
Conductor $2359296$
Sign $1$
Analytic cond. $150.430$
Root an. cond. $3.50214$
Motivic weight $1$
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  + 2·3-s + 4·5-s + 3·9-s − 4·11-s + 4·13-s + 8·15-s + 4·17-s + 8·23-s + 4·25-s + 4·27-s + 12·29-s − 8·33-s + 12·37-s + 8·39-s − 4·41-s + 8·43-s + 12·45-s + 8·47-s − 12·49-s + 8·51-s + 12·53-s − 16·55-s − 8·59-s + 12·61-s + 16·65-s + 16·69-s + 8·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s − 1.20·11-s + 1.10·13-s + 2.06·15-s + 0.970·17-s + 1.66·23-s + 4/5·25-s + 0.769·27-s + 2.22·29-s − 1.39·33-s + 1.97·37-s + 1.28·39-s − 0.624·41-s + 1.21·43-s + 1.78·45-s + 1.16·47-s − 1.71·49-s + 1.12·51-s + 1.64·53-s − 2.15·55-s − 1.04·59-s + 1.53·61-s + 1.98·65-s + 1.92·69-s + 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2359296\)    =    \(2^{18} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(150.430\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2359296,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.542148911\)
\(L(\frac12)\) \(\approx\) \(6.542148911\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_m
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_s
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_w
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_be
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_cc
29$D_{4}$ \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_do
31$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.31.a_m
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.37.am_da
41$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_o
43$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dq
47$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_bm
53$D_{4}$ \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_eu
59$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_dy
61$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_ew
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.67.a_dy
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.73.a_s
79$D_{4}$ \( 1 + 32 T + 412 T^{2} + 32 p T^{3} + p^{2} T^{4} \) 2.79.bg_pw
83$D_{4}$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hm
89$C_4$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_di
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gk
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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

−9.694468455169911339648426407487, −9.263247525888969910434106896920, −8.726242909827584481788428541958, −8.680088304946863075078983045293, −8.034177794536019404695437551618, −7.902143102458363450844213594938, −7.08914896736604951814659432982, −7.06217321280785590516066329506, −6.26083503226687910235809933465, −6.03927125108928665263293464136, −5.48865383006853673783369906253, −5.32388382876011597818427962489, −4.52854011525196972213481184143, −4.27645982070683424710230626549, −3.47215109372313747215187451736, −2.88428709462737857148108864879, −2.67676023609515521262976787307, −2.28089429096443038666811498393, −1.22361178559583339778249123595, −1.20272965852503917689697692874, 1.20272965852503917689697692874, 1.22361178559583339778249123595, 2.28089429096443038666811498393, 2.67676023609515521262976787307, 2.88428709462737857148108864879, 3.47215109372313747215187451736, 4.27645982070683424710230626549, 4.52854011525196972213481184143, 5.32388382876011597818427962489, 5.48865383006853673783369906253, 6.03927125108928665263293464136, 6.26083503226687910235809933465, 7.06217321280785590516066329506, 7.08914896736604951814659432982, 7.902143102458363450844213594938, 8.034177794536019404695437551618, 8.680088304946863075078983045293, 8.726242909827584481788428541958, 9.263247525888969910434106896920, 9.694468455169911339648426407487

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