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Properties

Label 4-648000-1.1-c1e2-0-0
Degree $4$
Conductor $648000$
Sign $1$
Analytic cond. $41.3170$
Root an. cond. $2.53531$
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  + 2-s − 4-s − 5-s − 3·8-s − 10-s − 4·13-s − 16-s + 20-s + 25-s − 4·26-s + 5·32-s − 20·37-s + 3·40-s − 20·41-s + 8·43-s − 14·49-s + 50-s + 4·52-s + 20·53-s + 7·64-s + 4·65-s + 24·67-s + 16·71-s − 20·74-s + 80-s − 20·82-s − 24·83-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.10·13-s − 1/4·16-s + 0.223·20-s + 1/5·25-s − 0.784·26-s + 0.883·32-s − 3.28·37-s + 0.474·40-s − 3.12·41-s + 1.21·43-s − 2·49-s + 0.141·50-s + 0.554·52-s + 2.74·53-s + 7/8·64-s + 0.496·65-s + 2.93·67-s + 1.89·71-s − 2.32·74-s + 0.111·80-s − 2.20·82-s − 2.63·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(648000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(41.3170\)
Root analytic conductor: \(2.53531\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 648000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.074326590\)
\(L(\frac12)\) \(\approx\) \(1.074326590\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.528796002193558283113883461082, −8.052642541080546717924291648317, −7.30265771947486168528941346448, −6.87126538623379444026486653759, −6.76955885158912340897756950945, −5.96744368889847453387068579979, −5.32790836897096555983732668084, −5.04860090093668311608273372623, −4.86275236111181745763302133865, −3.97486198573635904909121919320, −3.63412818236731287070311550143, −3.23564539976077866970720545176, −2.43462761639899639642195755230, −1.76953374967632241466455046617, −0.45394994076227052108962099324, 0.45394994076227052108962099324, 1.76953374967632241466455046617, 2.43462761639899639642195755230, 3.23564539976077866970720545176, 3.63412818236731287070311550143, 3.97486198573635904909121919320, 4.86275236111181745763302133865, 5.04860090093668311608273372623, 5.32790836897096555983732668084, 5.96744368889847453387068579979, 6.76955885158912340897756950945, 6.87126538623379444026486653759, 7.30265771947486168528941346448, 8.052642541080546717924291648317, 8.528796002193558283113883461082

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