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Properties

Label 4-9600e2-1.1-c1e2-0-18
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.c_ac
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_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.ac_w
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_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.g_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.e_w
47$D_{4}$ \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.47.as_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.am_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.e_cs
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_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.s_io
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_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

−7.19846810431574290981867476285, −7.13281676121508867763044149506, −6.86002199324058115059820036015, −6.55641183035520200721373544511, −6.12761243468705278214676347102, −5.75008567828915005919120496302, −5.63921457282558883263832434147, −5.22977205173193012918367569479, −4.57891546640600230140482305557, −4.35970901082863681374702336715, −4.28835590492216208117648642888, −3.70987066611989099916776199411, −3.13576665022591325405146330596, −3.00825300817284749679363240475, −2.28819710648595658225829157061, −1.89767570931191236621115787203, −1.35361922002254317026469649773, −0.946047332825395223855990091400, 0, 0, 0.946047332825395223855990091400, 1.35361922002254317026469649773, 1.89767570931191236621115787203, 2.28819710648595658225829157061, 3.00825300817284749679363240475, 3.13576665022591325405146330596, 3.70987066611989099916776199411, 4.28835590492216208117648642888, 4.35970901082863681374702336715, 4.57891546640600230140482305557, 5.22977205173193012918367569479, 5.63921457282558883263832434147, 5.75008567828915005919120496302, 6.12761243468705278214676347102, 6.55641183035520200721373544511, 6.86002199324058115059820036015, 7.13281676121508867763044149506, 7.19846810431574290981867476285

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