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Properties

Label 4-56e4-1.1-c1e2-0-17
Degree $4$
Conductor $9834496$
Sign $1$
Analytic cond. $627.055$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 9-s + 2·11-s + 4·15-s − 6·17-s − 10·19-s − 2·23-s + 25-s + 6·27-s + 14·31-s − 4·33-s − 6·37-s − 8·41-s − 8·43-s + 2·45-s + 18·47-s + 12·51-s + 2·53-s − 4·55-s + 20·57-s − 2·59-s − 6·61-s + 14·67-s + 4·69-s + 16·71-s − 18·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1/3·9-s + 0.603·11-s + 1.03·15-s − 1.45·17-s − 2.29·19-s − 0.417·23-s + 1/5·25-s + 1.15·27-s + 2.51·31-s − 0.696·33-s − 0.986·37-s − 1.24·41-s − 1.21·43-s + 0.298·45-s + 2.62·47-s + 1.68·51-s + 0.274·53-s − 0.539·55-s + 2.64·57-s − 0.260·59-s − 0.768·61-s + 1.71·67-s + 0.481·69-s + 1.89·71-s − 2.10·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9834496\)    =    \(2^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(627.055\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9834496,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_f
5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_d
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_v
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bj
19$D_{4}$ \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.19.k_cj
23$D_{4}$ \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bd
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$D_{4}$ \( 1 - 14 T + 109 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_ef
37$D_{4}$ \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_bz
41$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_dm
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 - 18 T + 173 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.47.as_gr
53$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.53.ac_ed
59$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_v
61$D_{4}$ \( 1 + 6 T + 99 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_dv
67$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_gj
71$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.71.aq_gs
73$D_{4}$ \( 1 + 18 T + 195 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.73.s_hn
79$D_{4}$ \( 1 + 2 T + 109 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_ef
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_cc
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$D_{4}$ \( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hu
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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

−8.525869511495533364063787678443, −8.456755405680584718158436550752, −7.65210829654827355902817645938, −7.37845521750453703711904354387, −6.76854708126124718990297845499, −6.55845099979017514652587119119, −6.19294842713957101226758428952, −6.14781680915652749414094951160, −5.29532961046423438129161658160, −5.15741418476605100511842224937, −4.44019108543844189953525274728, −4.36590429000383657887290331704, −3.92856865693544620443884369723, −3.45835995508832654635027905907, −2.65669433272038829010243279193, −2.43269130072917020933100561140, −1.75193485107840393676975194688, −0.951957699095178357635460604610, 0, 0, 0.951957699095178357635460604610, 1.75193485107840393676975194688, 2.43269130072917020933100561140, 2.65669433272038829010243279193, 3.45835995508832654635027905907, 3.92856865693544620443884369723, 4.36590429000383657887290331704, 4.44019108543844189953525274728, 5.15741418476605100511842224937, 5.29532961046423438129161658160, 6.14781680915652749414094951160, 6.19294842713957101226758428952, 6.55845099979017514652587119119, 6.76854708126124718990297845499, 7.37845521750453703711904354387, 7.65210829654827355902817645938, 8.456755405680584718158436550752, 8.525869511495533364063787678443

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