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Properties

Label 4-567675-1.1-c1e2-0-3
Degree $4$
Conductor $567675$
Sign $1$
Analytic cond. $36.1954$
Root an. cond. $2.45280$
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  + 3-s − 3·4-s + 8·7-s + 9-s − 3·12-s + 12·13-s + 5·16-s − 8·19-s + 8·21-s + 25-s + 27-s − 24·28-s − 16·31-s − 3·36-s + 4·37-s + 12·39-s + 8·43-s + 5·48-s + 34·49-s − 36·52-s − 8·57-s − 20·61-s + 8·63-s − 3·64-s + 16·67-s − 4·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 3.02·7-s + 1/3·9-s − 0.866·12-s + 3.32·13-s + 5/4·16-s − 1.83·19-s + 1.74·21-s + 1/5·25-s + 0.192·27-s − 4.53·28-s − 2.87·31-s − 1/2·36-s + 0.657·37-s + 1.92·39-s + 1.21·43-s + 0.721·48-s + 34/7·49-s − 4.99·52-s − 1.05·57-s − 2.56·61-s + 1.00·63-s − 3/8·64-s + 1.95·67-s − 0.468·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(567675\)    =    \(3^{3} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(36.1954\)
Root analytic conductor: \(2.45280\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 567675,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.871498698\)
\(L(\frac12)\) \(\approx\) \(2.871498698\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
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

−8.346606260262100304740118719484, −8.285142698379180989401958892533, −7.82152461814686983756055018114, −7.41886368904450739537189724511, −6.53264436370468127763928142197, −6.04599499864174651869297996238, −5.35775714661900019913688256525, −5.30506594738975870887892595526, −4.31801287744868728937060903572, −4.23882127299780961454300589319, −3.97331207281193943578109005786, −3.22174431616232965676904645612, −1.93438057192777317348195684356, −1.68581285052697885373760198065, −0.984285408377975149151993884914, 0.984285408377975149151993884914, 1.68581285052697885373760198065, 1.93438057192777317348195684356, 3.22174431616232965676904645612, 3.97331207281193943578109005786, 4.23882127299780961454300589319, 4.31801287744868728937060903572, 5.30506594738975870887892595526, 5.35775714661900019913688256525, 6.04599499864174651869297996238, 6.53264436370468127763928142197, 7.41886368904450739537189724511, 7.82152461814686983756055018114, 8.285142698379180989401958892533, 8.346606260262100304740118719484

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