Properties

Label 4-4608e2-1.1-c1e2-0-23
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 10·13-s − 16·17-s + 2·25-s + 6·29-s + 14·37-s + 14·49-s + 18·53-s + 22·61-s + 20·65-s + 32·85-s − 16·97-s + 18·101-s + 26·109-s + 28·113-s − 10·125-s + ⋯
L(s)  = 1  − 0.894·5-s − 2.77·13-s − 3.88·17-s + 2/5·25-s + 1.11·29-s + 2.30·37-s + 2·49-s + 2.47·53-s + 2.81·61-s + 2.48·65-s + 3.47·85-s − 1.62·97-s + 1.79·101-s + 2.49·109-s + 2.63·113-s − 0.894·125-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

\(L(1)\) \(\approx\) \(1.362669981\)
\(L(\frac12)\) \(\approx\) \(1.362669981\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.c_c
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.k_by
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_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.ag_s
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ao_du
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.as_gg
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.61.aw_ji
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.q_jy
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.385479279033368861609707424836, −8.383029175909405037583411973201, −7.48857037009772824668698172985, −7.40083421460210234008308722372, −7.18257422426829392622792156329, −6.83734430494136487379030723407, −6.27620456058626321556375541378, −6.23343047959648399832907181062, −5.36295181086017796248768573416, −5.11104595264915916438082198079, −4.65457869179857705037560389834, −4.41248316446448755597460755748, −4.09309636903869768195423940014, −3.83672727783151552714460959956, −2.81850783061196489178662367365, −2.59231654955138619059334936847, −2.26802468492041229284588118948, −2.05254973620759833811881740968, −0.63392393671241026761669066715, −0.51975964016386229079100706753, 0.51975964016386229079100706753, 0.63392393671241026761669066715, 2.05254973620759833811881740968, 2.26802468492041229284588118948, 2.59231654955138619059334936847, 2.81850783061196489178662367365, 3.83672727783151552714460959956, 4.09309636903869768195423940014, 4.41248316446448755597460755748, 4.65457869179857705037560389834, 5.11104595264915916438082198079, 5.36295181086017796248768573416, 6.23343047959648399832907181062, 6.27620456058626321556375541378, 6.83734430494136487379030723407, 7.18257422426829392622792156329, 7.40083421460210234008308722372, 7.48857037009772824668698172985, 8.383029175909405037583411973201, 8.385479279033368861609707424836

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