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Properties

Label 4-720e2-1.1-c1e2-0-39
Degree $4$
Conductor $518400$
Sign $1$
Analytic cond. $33.0536$
Root an. cond. $2.39775$
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  + 2·2-s + 2·4-s − 2·5-s − 4·10-s + 6·11-s + 6·13-s − 4·16-s − 2·19-s − 4·20-s + 12·22-s + 16·23-s − 25-s + 12·26-s − 6·29-s − 8·32-s + 6·37-s − 4·38-s + 6·43-s + 12·44-s + 32·46-s − 14·49-s − 2·50-s + 12·52-s + 18·53-s − 12·55-s − 12·58-s − 18·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s + 1.80·11-s + 1.66·13-s − 16-s − 0.458·19-s − 0.894·20-s + 2.55·22-s + 3.33·23-s − 1/5·25-s + 2.35·26-s − 1.11·29-s − 1.41·32-s + 0.986·37-s − 0.648·38-s + 0.914·43-s + 1.80·44-s + 4.71·46-s − 2·49-s − 0.282·50-s + 1.66·52-s + 2.47·53-s − 1.61·55-s − 1.57·58-s − 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(33.0536\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.216839901\)
\(L(\frac12)\) \(\approx\) \(4.216839901\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_s
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_s
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_c
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.23.aq_eg
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.g_s
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_s
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_s
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.47.a_adm
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.as_gg
59$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_gg
61$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_by
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_s
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_gg
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−10.92023239810247055995245921225, −10.54762015453722569392200215276, −9.485014680690147277290111964959, −9.305858781316963035859469230036, −8.891372493050236921324771493443, −8.652850045205801632123609314167, −7.956046592461271924806820942653, −7.37932400712552283860942124974, −7.01385121640649142756975729189, −6.45278279873154311563427600862, −6.24776571890206853262596818762, −5.70991505358925597335335818308, −5.12062100164718091409456953175, −4.45473078778446469776359931836, −4.27164574432632001134687497610, −3.58170783695403999621274150417, −3.40861988712815530560269136261, −2.79129162062936904829559365960, −1.67844278156686471158517366421, −0.946333878157062641540079566239, 0.946333878157062641540079566239, 1.67844278156686471158517366421, 2.79129162062936904829559365960, 3.40861988712815530560269136261, 3.58170783695403999621274150417, 4.27164574432632001134687497610, 4.45473078778446469776359931836, 5.12062100164718091409456953175, 5.70991505358925597335335818308, 6.24776571890206853262596818762, 6.45278279873154311563427600862, 7.01385121640649142756975729189, 7.37932400712552283860942124974, 7.956046592461271924806820942653, 8.652850045205801632123609314167, 8.891372493050236921324771493443, 9.305858781316963035859469230036, 9.485014680690147277290111964959, 10.54762015453722569392200215276, 10.92023239810247055995245921225

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