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Properties

Label 4-9600e2-1.1-c1e2-0-28
Degree $4$
Conductor $92160000$
Sign $1$
Analytic cond. $5876.20$
Root an. cond. $8.75536$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s + 3·9-s − 4·11-s − 2·13-s − 6·17-s − 2·19-s + 4·21-s − 2·23-s + 4·27-s + 6·31-s − 8·33-s + 2·37-s − 4·39-s − 4·41-s + 4·43-s − 18·47-s + 6·49-s − 12·51-s − 20·53-s − 4·57-s − 12·59-s − 4·61-s + 6·63-s + 4·67-s − 4·69-s + 16·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s + 0.872·21-s − 0.417·23-s + 0.769·27-s + 1.07·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 0.624·41-s + 0.609·43-s − 2.62·47-s + 6/7·49-s − 1.68·51-s − 2.74·53-s − 0.529·57-s − 1.56·59-s − 0.512·61-s + 0.755·63-s + 0.488·67-s − 0.481·69-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92160000\)    =    \(2^{14} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5876.20\)
Root analytic conductor: \(8.75536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 92160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ac
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_k
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_ba
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_w
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_be
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$D_{4}$ \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cc
37$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_cg
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_w
47$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.47.s_gc
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_cs
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_dq
79$D_{4}$ \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.79.as_io
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−7.68000445049182951241916863412, −7.29258196287385806982992246008, −6.82216966543961239273177646666, −6.58937695543102705965759717968, −6.26306015914245922683108662934, −5.91044808789046285880545403631, −5.22482614286641376420958949185, −4.98285669183716440274428081484, −4.71794775404622438084136123861, −4.49932585098856861215033379606, −3.96122367147727515843844431394, −3.67054016579666201725284017080, −2.99423792522344929367161348330, −2.83438031636290181879678904414, −2.41752823443718083830712036522, −2.08206522742949308385468259978, −1.57809938123855289726341672185, −1.23548665275784589508844217394, 0, 0, 1.23548665275784589508844217394, 1.57809938123855289726341672185, 2.08206522742949308385468259978, 2.41752823443718083830712036522, 2.83438031636290181879678904414, 2.99423792522344929367161348330, 3.67054016579666201725284017080, 3.96122367147727515843844431394, 4.49932585098856861215033379606, 4.71794775404622438084136123861, 4.98285669183716440274428081484, 5.22482614286641376420958949185, 5.91044808789046285880545403631, 6.26306015914245922683108662934, 6.58937695543102705965759717968, 6.82216966543961239273177646666, 7.29258196287385806982992246008, 7.68000445049182951241916863412

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