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Properties

Label 4-1512e2-1.1-c1e2-0-22
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  + 4·5-s + 2·7-s + 8·17-s + 2·19-s + 4·23-s + 5·25-s + 4·29-s − 10·31-s + 8·35-s + 6·37-s + 8·41-s + 4·43-s + 4·47-s + 3·49-s + 16·53-s + 12·59-s + 8·61-s − 24·67-s + 12·71-s − 4·73-s − 4·79-s − 12·83-s + 32·85-s + 32·89-s + 8·95-s − 16·97-s − 18·103-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.755·7-s + 1.94·17-s + 0.458·19-s + 0.834·23-s + 25-s + 0.742·29-s − 1.79·31-s + 1.35·35-s + 0.986·37-s + 1.24·41-s + 0.609·43-s + 0.583·47-s + 3/7·49-s + 2.19·53-s + 1.56·59-s + 1.02·61-s − 2.93·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s − 1.31·83-s + 3.47·85-s + 3.39·89-s + 0.820·95-s − 1.62·97-s − 1.77·103-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

\(L(1)\) \(\approx\) \(4.790611056\)
\(L(\frac12)\) \(\approx\) \(4.790611056\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$D_{4}$ \( 1 - 2 T - 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_aj
23$D_{4}$ \( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bv
29$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_o
31$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_cx
37$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ct
41$D_{4}$ \( 1 - 8 T + 71 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_ct
43$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_da
47$D_{4}$ \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_di
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fm
61$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_dm
67$D_{4}$ \( 1 + 24 T + 266 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.67.y_kg
71$D_{4}$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_dz
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_bq
79$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_cc
83$D_{4}$ \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_fy
89$D_{4}$ \( 1 - 32 T + 431 T^{2} - 32 p T^{3} + p^{2} T^{4} \) 2.89.abg_qp
97$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.97.q_ic
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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

−9.621546737144327897871465773623, −9.366050569748867859493408661165, −8.848174497036033205657547335538, −8.739699860124162947939662373389, −7.955069964060904614450729302240, −7.69350774088941189907911998352, −7.22951191492484334055534496009, −7.00397195988806834503008826068, −6.15667459762770627374079003275, −5.94032351379157600151098219327, −5.52766286654012376560561691630, −5.32274104793998930414910573708, −4.88308638061677817043011963480, −4.12045328296947216208219627951, −3.75331969499395054947676936677, −3.04481976939573819137472147040, −2.42803383589021507711138740486, −2.17858273755822238038639113327, −1.17236735730366141167327305531, −1.11494321412766035622523018944, 1.11494321412766035622523018944, 1.17236735730366141167327305531, 2.17858273755822238038639113327, 2.42803383589021507711138740486, 3.04481976939573819137472147040, 3.75331969499395054947676936677, 4.12045328296947216208219627951, 4.88308638061677817043011963480, 5.32274104793998930414910573708, 5.52766286654012376560561691630, 5.94032351379157600151098219327, 6.15667459762770627374079003275, 7.00397195988806834503008826068, 7.22951191492484334055534496009, 7.69350774088941189907911998352, 7.955069964060904614450729302240, 8.739699860124162947939662373389, 8.848174497036033205657547335538, 9.366050569748867859493408661165, 9.621546737144327897871465773623

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