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Properties

Label 4-4032e2-1.1-c1e2-0-12
Degree $4$
Conductor $16257024$
Sign $1$
Analytic cond. $1036.56$
Root an. cond. $5.67412$
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 − 2·7-s + 4·11-s − 6·13-s + 2·19-s + 8·23-s − 2·25-s − 4·31-s − 4·35-s + 8·41-s + 4·43-s + 12·47-s + 3·49-s − 20·53-s + 8·55-s + 14·59-s − 18·61-s − 12·65-s + 8·67-s + 8·71-s + 12·73-s − 8·77-s − 8·79-s + 14·83-s + 12·89-s + 12·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s + 1.20·11-s − 1.66·13-s + 0.458·19-s + 1.66·23-s − 2/5·25-s − 0.718·31-s − 0.676·35-s + 1.24·41-s + 0.609·43-s + 1.75·47-s + 3/7·49-s − 2.74·53-s + 1.07·55-s + 1.82·59-s − 2.30·61-s − 1.48·65-s + 0.977·67-s + 0.949·71-s + 1.40·73-s − 0.911·77-s − 0.900·79-s + 1.53·83-s + 1.27·89-s + 1.25·91-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1036.56\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16257024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.196119673\)
\(L(\frac12)\) \(\approx\) \(3.196119673\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_g
11$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_g
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_be
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$D_{4}$ \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bi
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bu
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.37.a_cc
41$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_da
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_cs
47$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_eg
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$D_{4}$ \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_gg
61$D_{4}$ \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_hq
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_da
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_dy
79$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_dq
83$D_{4}$ \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.83.ao_ic
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$D_{4}$ \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_je
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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

−8.920868537315266251190266548871, −8.274456356665318971536017188428, −7.59441071030179057448961562297, −7.55697032260175243964962435700, −7.24231808989139700875491340724, −6.75194237814926577108489139463, −6.26906195584521488628022417148, −6.23801249531296123197739089397, −5.73137956258066710161386631477, −5.19792256139613805397175975375, −4.94741979568560377917040024331, −4.60349033228626955811018317994, −3.94941673211153744495821271853, −3.65448366850550441181475000701, −3.07854810057956205889283035085, −2.77162877342586045691452326091, −2.07088810042830686614321793020, −1.96462480003166355943015198923, −1.01446912297689590772616054098, −0.59114936311307336238315290528, 0.59114936311307336238315290528, 1.01446912297689590772616054098, 1.96462480003166355943015198923, 2.07088810042830686614321793020, 2.77162877342586045691452326091, 3.07854810057956205889283035085, 3.65448366850550441181475000701, 3.94941673211153744495821271853, 4.60349033228626955811018317994, 4.94741979568560377917040024331, 5.19792256139613805397175975375, 5.73137956258066710161386631477, 6.23801249531296123197739089397, 6.26906195584521488628022417148, 6.75194237814926577108489139463, 7.24231808989139700875491340724, 7.55697032260175243964962435700, 7.59441071030179057448961562297, 8.274456356665318971536017188428, 8.920868537315266251190266548871

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