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Properties

Label 4-9600e2-1.1-c1e2-0-22
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 − 2·13-s + 4·17-s + 2·19-s + 4·21-s − 8·23-s − 4·27-s + 4·29-s − 6·31-s + 4·39-s + 8·41-s + 2·43-s + 4·47-s − 49-s − 8·51-s − 8·53-s − 4·57-s − 4·59-s − 2·61-s − 6·63-s − 18·67-s + 16·69-s + 4·71-s + 8·73-s + 16·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 9-s − 0.554·13-s + 0.970·17-s + 0.458·19-s + 0.872·21-s − 1.66·23-s − 0.769·27-s + 0.742·29-s − 1.07·31-s + 0.640·39-s + 1.24·41-s + 0.304·43-s + 0.583·47-s − 1/7·49-s − 1.12·51-s − 1.09·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.755·63-s − 2.19·67-s + 1.92·69-s + 0.474·71-s + 0.936·73-s + 1.80·79-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$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_f
11$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.11.a_m
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$D_{4}$ \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_bc
19$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bd
23$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_ca
29$D_{4}$ \( 1 - 4 T + 52 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_ca
31$D_{4}$ \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cj
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_dk
43$D_{4}$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_ad
47$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_cg
53$D_{4}$ \( 1 + 8 T + 112 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_ei
59$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_de
61$D_{4}$ \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_df
67$D_{4}$ \( 1 + 18 T + 205 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.67.s_hx
71$D_{4}$ \( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_ce
73$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_es
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_fa
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.97.abe_qd
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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

−7.55314917036186562279127505773, −7.20411356671769848858591746947, −6.60109346500775357542580631770, −6.56108766046864362356893339957, −5.99409666668898123423408897635, −5.95962549256733145719879233697, −5.55977029047781838118132644655, −5.08233808679202032515585903423, −4.90718210565905267196005970172, −4.43238070094134476033342769593, −3.89613771997880241180397471468, −3.85897254001055051405499786556, −3.13439102668219766381245251861, −3.01179425822505994295508094816, −2.19763463475141353318655225527, −2.07597657471280961452740394339, −1.19930999088746990079989454028, −1.01761472398431181520397850789, 0, 0, 1.01761472398431181520397850789, 1.19930999088746990079989454028, 2.07597657471280961452740394339, 2.19763463475141353318655225527, 3.01179425822505994295508094816, 3.13439102668219766381245251861, 3.85897254001055051405499786556, 3.89613771997880241180397471468, 4.43238070094134476033342769593, 4.90718210565905267196005970172, 5.08233808679202032515585903423, 5.55977029047781838118132644655, 5.95962549256733145719879233697, 5.99409666668898123423408897635, 6.56108766046864362356893339957, 6.60109346500775357542580631770, 7.20411356671769848858591746947, 7.55314917036186562279127505773

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