Properties

Label 4-4608e2-1.1-c1e2-0-43
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
Motivic weight $1$
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  + 6·5-s − 2·13-s + 16·17-s + 18·25-s + 14·29-s − 10·37-s + 14·49-s + 10·53-s − 2·61-s − 12·65-s + 96·85-s + 16·97-s − 22·101-s − 14·109-s + 28·113-s + 30·125-s + ⋯
L(s)  = 1  + 2.68·5-s − 0.554·13-s + 3.88·17-s + 18/5·25-s + 2.59·29-s − 1.64·37-s + 2·49-s + 1.37·53-s − 0.256·61-s − 1.48·65-s + 10.4·85-s + 1.62·97-s − 2.18·101-s − 1.34·109-s + 2.63·113-s + 2.68·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.708425478\)
\(L(\frac12)\) \(\approx\) \(7.708425478\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.5.ag_s
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + p^{2} T^{4} \) 2.11.a_a
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.17.aq_du
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.29.ao_du
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ak_by
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.a_ada
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−8.444339767221720815065690974557, −8.415649201722094424648853987817, −7.53487413581473425483713159412, −7.51719143075904109050504368489, −7.01429185871357887497001859994, −6.63674038702804376202000820875, −6.12646515190230989904541770405, −5.79673115159510924506174080134, −5.73032623966953883786585968624, −5.31185354886742453308347673672, −4.91184928332677817489835289825, −4.74894614158047563019876578007, −3.71876353944584583120416278258, −3.59548909589004859751465859047, −2.90864452732072840110639155326, −2.60111374650533563917678838096, −2.27784378579788762081068248273, −1.55038604139838180383072939616, −1.20788235809756206994827948560, −0.856904389534332756544661305871, 0.856904389534332756544661305871, 1.20788235809756206994827948560, 1.55038604139838180383072939616, 2.27784378579788762081068248273, 2.60111374650533563917678838096, 2.90864452732072840110639155326, 3.59548909589004859751465859047, 3.71876353944584583120416278258, 4.74894614158047563019876578007, 4.91184928332677817489835289825, 5.31185354886742453308347673672, 5.73032623966953883786585968624, 5.79673115159510924506174080134, 6.12646515190230989904541770405, 6.63674038702804376202000820875, 7.01429185871357887497001859994, 7.51719143075904109050504368489, 7.53487413581473425483713159412, 8.415649201722094424648853987817, 8.444339767221720815065690974557

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