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Properties

Label 4-920e2-1.1-c1e2-0-17
Degree $4$
Conductor $846400$
Sign $1$
Analytic cond. $53.9671$
Root an. cond. $2.71039$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 7-s − 9-s − 4·11-s − 3·13-s + 2·15-s + 7·17-s − 4·19-s + 21-s − 2·23-s + 3·25-s − 8·29-s − 8·31-s + 4·33-s + 2·35-s − 3·37-s + 3·39-s − 4·43-s + 2·45-s − 5·47-s − 9·49-s − 7·51-s + 13·53-s + 8·55-s + 4·57-s − 7·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s − 1/3·9-s − 1.20·11-s − 0.832·13-s + 0.516·15-s + 1.69·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s + 3/5·25-s − 1.48·29-s − 1.43·31-s + 0.696·33-s + 0.338·35-s − 0.493·37-s + 0.480·39-s − 0.609·43-s + 0.298·45-s − 0.729·47-s − 9/7·49-s − 0.980·51-s + 1.78·53-s + 1.07·55-s + 0.529·57-s − 0.911·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(846400\)    =    \(2^{6} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(53.9671\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 846400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
7$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_k
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_y
17$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bq
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
29$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cf
31$D_{4}$ \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cj
37$D_{4}$ \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_cu
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.41.a_cn
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_w
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_ck
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.53.an_fo
59$D_{4}$ \( 1 + 7 T + 126 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.59.h_ew
61$D_{4}$ \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_fa
67$D_{4}$ \( 1 + 19 T + 220 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.67.t_im
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.71.k_gl
73$D_{4}$ \( 1 - 11 T + 172 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_gq
79$D_{4}$ \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gk
83$D_{4}$ \( 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_fa
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.89.aq_ji
97$D_{4}$ \( 1 - 2 T + 178 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_gw
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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

−10.04235762469178362002793081819, −9.527700492840389776085760445404, −8.918055005191489921962297743259, −8.744167355256539290499300744553, −7.87787094055281652610846232806, −7.82217264057467387909205067532, −7.41800561273076131734096444241, −7.08046810530948295305554100968, −6.26167348608412767415468072335, −6.00307995101330595635695902458, −5.32244973383612532078712401436, −5.25635349558868339763553177976, −4.58372953865440310153755007996, −3.99454410774820647075962821105, −3.34803918581233743638264399853, −3.11920822435374775943149076422, −2.27445453388986391326208563508, −1.54170528812686804589089880290, 0, 0, 1.54170528812686804589089880290, 2.27445453388986391326208563508, 3.11920822435374775943149076422, 3.34803918581233743638264399853, 3.99454410774820647075962821105, 4.58372953865440310153755007996, 5.25635349558868339763553177976, 5.32244973383612532078712401436, 6.00307995101330595635695902458, 6.26167348608412767415468072335, 7.08046810530948295305554100968, 7.41800561273076131734096444241, 7.82217264057467387909205067532, 7.87787094055281652610846232806, 8.744167355256539290499300744553, 8.918055005191489921962297743259, 9.527700492840389776085760445404, 10.04235762469178362002793081819

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