Properties

Label 4-1776e2-1.1-c1e2-0-5
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $201.112$
Root an. cond. $3.76582$
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 − 3·7-s + 3·9-s − 3·11-s + 13-s + 5·17-s + 3·19-s + 6·21-s − 9·23-s − 10·25-s − 4·27-s + 16·29-s + 6·31-s + 6·33-s + 2·37-s − 2·39-s + 4·41-s − 8·43-s + 10·47-s + 3·49-s − 10·51-s + 7·53-s − 6·57-s + 6·59-s + 12·61-s − 9·63-s + 18·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.13·7-s + 9-s − 0.904·11-s + 0.277·13-s + 1.21·17-s + 0.688·19-s + 1.30·21-s − 1.87·23-s − 2·25-s − 0.769·27-s + 2.97·29-s + 1.07·31-s + 1.04·33-s + 0.328·37-s − 0.320·39-s + 0.624·41-s − 1.21·43-s + 1.45·47-s + 3/7·49-s − 1.40·51-s + 0.961·53-s − 0.794·57-s + 0.781·59-s + 1.53·61-s − 1.13·63-s + 2.19·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(201.112\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3154176,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.172600749\)
\(L(\frac12)\) \(\approx\) \(1.172600749\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_g
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_o
13$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_q
17$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_be
19$D_{4}$ \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_be
23$D_{4}$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_ce
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_be
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_da
53$D_{4}$ \( 1 - 7 T + 108 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.53.ah_ee
59$D_{4}$ \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_di
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$D_{4}$ \( 1 - 18 T + 174 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.67.as_gs
71$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_as
73$C_2^2$ \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.73.n_ds
79$D_{4}$ \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_eo
83$D_{4}$ \( 1 - 13 T + 198 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.83.an_hq
89$D_{4}$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.89.ab_ada
97$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_bu
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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

−9.722887655888399932317502225141, −9.369479645306902466163058794542, −8.461829906023158474795962782327, −8.290828059306001851630845225857, −7.896503725508138224297692897154, −7.57878596142603729807709001299, −6.82898785232762015612828845703, −6.77535372033194070558238824285, −6.08269637991764331603323137355, −5.99483416243670898925189391089, −5.38680539567046088149721017015, −5.34444473260172080291025242396, −4.41567813981223773172450946413, −4.25499390031747947366584582446, −3.60214469778265190561992665615, −3.18490113056377681171550381060, −2.51975725437602115827676653242, −2.04838815191022239326953273883, −0.969288985420844638553990087393, −0.55544463708965224718689491760, 0.55544463708965224718689491760, 0.969288985420844638553990087393, 2.04838815191022239326953273883, 2.51975725437602115827676653242, 3.18490113056377681171550381060, 3.60214469778265190561992665615, 4.25499390031747947366584582446, 4.41567813981223773172450946413, 5.34444473260172080291025242396, 5.38680539567046088149721017015, 5.99483416243670898925189391089, 6.08269637991764331603323137355, 6.77535372033194070558238824285, 6.82898785232762015612828845703, 7.57878596142603729807709001299, 7.896503725508138224297692897154, 8.290828059306001851630845225857, 8.461829906023158474795962782327, 9.369479645306902466163058794542, 9.722887655888399932317502225141

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