Properties

Label 4-40e4-1.1-c1e2-0-36
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  − 9-s + 8·13-s + 14·17-s + 4·37-s + 10·41-s + 6·49-s + 12·53-s − 20·61-s + 18·73-s − 8·81-s − 10·89-s − 4·97-s + 4·101-s − 12·109-s + 2·113-s − 8·117-s − 17·121-s + ⋯
L(s)  = 1  − 1/3·9-s + 2.21·13-s + 3.39·17-s + 0.657·37-s + 1.56·41-s + 6/7·49-s + 1.64·53-s − 2.56·61-s + 2.10·73-s − 8/9·81-s − 1.05·89-s − 0.406·97-s + 0.398·101-s − 1.14·109-s + 0.188·113-s − 0.739·117-s − 1.54·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.449777640\)
\(L(\frac12)\) \(\approx\) \(3.449777640\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.17.ao_df
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.19.a_ah
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.31.a_bq
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.41.ak_ed
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 + 129 T^{2} + p^{2} T^{4} \) 2.67.a_ez
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \) 2.79.a_fi
83$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.83.a_bp
89$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.89.k_hv
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−9.474626111714987005105452428289, −9.367359995447237594093830192117, −8.752116593395323242801923147869, −8.351852504600398419862760046948, −7.996825105458390711585015727874, −7.77361375731057492250159914064, −7.29576770062729789200049563035, −6.81597589739661095394337101429, −6.13240985028103065463642227541, −5.97216720530962660687726711371, −5.47532704787015703891955995501, −5.44232202253705255954242354875, −4.52793571040102260089164909665, −3.95258882913647270386748568644, −3.73742767373973948123899984573, −3.04418283685822809452542412477, −2.92892832482561331793376992509, −1.90743131121157376759686576618, −1.07252425096184911019067718479, −0.971631826011971749588523265304, 0.971631826011971749588523265304, 1.07252425096184911019067718479, 1.90743131121157376759686576618, 2.92892832482561331793376992509, 3.04418283685822809452542412477, 3.73742767373973948123899984573, 3.95258882913647270386748568644, 4.52793571040102260089164909665, 5.44232202253705255954242354875, 5.47532704787015703891955995501, 5.97216720530962660687726711371, 6.13240985028103065463642227541, 6.81597589739661095394337101429, 7.29576770062729789200049563035, 7.77361375731057492250159914064, 7.996825105458390711585015727874, 8.351852504600398419862760046948, 8.752116593395323242801923147869, 9.367359995447237594093830192117, 9.474626111714987005105452428289

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