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Properties

Label 4-3024e2-1.1-c1e2-0-20
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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  − 3·5-s + 4·7-s − 3·11-s − 9·13-s + 8·19-s − 9·23-s + 25-s − 9·29-s + 7·31-s − 12·35-s + 20·37-s + 9·41-s + 15·43-s + 3·47-s + 9·49-s + 12·53-s + 9·55-s − 3·59-s + 3·61-s + 27·65-s − 15·67-s − 12·77-s − 9·79-s − 9·83-s − 36·91-s − 24·95-s − 9·97-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.51·7-s − 0.904·11-s − 2.49·13-s + 1.83·19-s − 1.87·23-s + 1/5·25-s − 1.67·29-s + 1.25·31-s − 2.02·35-s + 3.28·37-s + 1.40·41-s + 2.28·43-s + 0.437·47-s + 9/7·49-s + 1.64·53-s + 1.21·55-s − 0.390·59-s + 0.384·61-s + 3.34·65-s − 1.83·67-s − 1.36·77-s − 1.01·79-s − 0.987·83-s − 3.77·91-s − 2.46·95-s − 0.913·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.599114411\)
\(L(\frac12)\) \(\approx\) \(1.599114411\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_o
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.j_bo
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_by
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_cq
43$C_2^2$ \( 1 - 15 T + 118 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_eo
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_cm
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_fm
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.73.a_adu
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.j_ec
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.j_eu
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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

−8.888768265293725083732454664789, −8.302729191105383647612976557044, −7.85497919882213340020942777364, −7.72796943325998890394483650937, −7.54357897384892909229362326162, −7.44335153854105164270703181236, −6.98567178857265511763919565286, −5.96916462689394014751658903153, −5.71614347182852355385653880975, −5.62374277322261289421656629045, −4.92882649822345887598498337339, −4.46794179651570461098117287457, −4.22071927690533667054716967124, −4.20166356256392410374852281644, −3.23536125241352348966895283713, −2.73579161567964325173198713219, −2.35161924972261865074890545498, −2.02042231894388616368019642682, −0.974872453086151160672866459934, −0.47796757664472220941816864649, 0.47796757664472220941816864649, 0.974872453086151160672866459934, 2.02042231894388616368019642682, 2.35161924972261865074890545498, 2.73579161567964325173198713219, 3.23536125241352348966895283713, 4.20166356256392410374852281644, 4.22071927690533667054716967124, 4.46794179651570461098117287457, 4.92882649822345887598498337339, 5.62374277322261289421656629045, 5.71614347182852355385653880975, 5.96916462689394014751658903153, 6.98567178857265511763919565286, 7.44335153854105164270703181236, 7.54357897384892909229362326162, 7.72796943325998890394483650937, 7.85497919882213340020942777364, 8.302729191105383647612976557044, 8.888768265293725083732454664789

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