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Properties

Label 4-1512e2-1.1-c1e2-0-8
Degree $4$
Conductor $2286144$
Sign $1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s + 4·11-s − 2·13-s + 4·19-s − 6·23-s + 5·25-s − 5·31-s − 8·35-s − 37-s − 8·41-s − 2·43-s − 4·47-s + 9·49-s + 6·53-s − 8·55-s + 3·61-s + 4·65-s − 11·67-s − 28·71-s + 14·73-s + 16·77-s − 13·79-s + 28·83-s − 6·89-s − 8·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 1.20·11-s − 0.554·13-s + 0.917·19-s − 1.25·23-s + 25-s − 0.898·31-s − 1.35·35-s − 0.164·37-s − 1.24·41-s − 0.304·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 1.07·55-s + 0.384·61-s + 0.496·65-s − 1.34·67-s − 3.32·71-s + 1.63·73-s + 1.82·77-s − 1.46·79-s + 3.07·83-s − 0.635·89-s − 0.838·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.085582321\)
\(L(\frac12)\) \(\approx\) \(2.085582321\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_ag
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_abf
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_aca
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.71.bc_na
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_et
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.n_dm
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.83.abc_ny
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.97.as_kp
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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

−9.598965139400811704567752916848, −9.063825146922620725638963314174, −8.951297834981584285047186033021, −8.451468426737524779257643468464, −7.928349240166063164959685697382, −7.79454727188045230766938865850, −7.38349497597691632079330980808, −6.86659241874847083867447749607, −6.63363186334519995382773677609, −5.87709095707091569619648553349, −5.55478974079256793713211384708, −4.93163193188680856381261427790, −4.71741545567992137796134865458, −4.18497595772537515592320783777, −3.75952702627859515378801027724, −3.32394498754455384129707769689, −2.65606180408635787200687168914, −1.71275475893907579126943292192, −1.62597588473604705754291005042, −0.58954452528650424285732341348, 0.58954452528650424285732341348, 1.62597588473604705754291005042, 1.71275475893907579126943292192, 2.65606180408635787200687168914, 3.32394498754455384129707769689, 3.75952702627859515378801027724, 4.18497595772537515592320783777, 4.71741545567992137796134865458, 4.93163193188680856381261427790, 5.55478974079256793713211384708, 5.87709095707091569619648553349, 6.63363186334519995382773677609, 6.86659241874847083867447749607, 7.38349497597691632079330980808, 7.79454727188045230766938865850, 7.928349240166063164959685697382, 8.451468426737524779257643468464, 8.951297834981584285047186033021, 9.063825146922620725638963314174, 9.598965139400811704567752916848

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