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Properties

Label 4-2400e2-1.1-c1e2-0-26
Degree $4$
Conductor $5760000$
Sign $1$
Analytic cond. $367.262$
Root an. cond. $4.37768$
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  − 9-s − 8·11-s − 8·19-s − 4·29-s − 8·31-s + 4·41-s − 2·49-s − 8·59-s + 12·61-s + 32·71-s + 8·79-s + 81-s − 20·89-s + 8·99-s − 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 2.41·11-s − 1.83·19-s − 0.742·29-s − 1.43·31-s + 0.624·41-s − 2/7·49-s − 1.04·59-s + 1.53·61-s + 3.79·71-s + 0.900·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s − 1.19·101-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5760000\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(367.262\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5760000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.71.abg_pi
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.97.a_c
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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

−8.758013295792964580189445896610, −8.175334856283479565184366458135, −8.031608597980090646472922657602, −7.86098078024792670901306255406, −7.18429845183093739347714495190, −6.98253808718326664738695357366, −6.25933822836416351247379396935, −6.20534235550408897063443770371, −5.36560366892489293594598388830, −5.28330711136303118673229284591, −5.05766962892158957405227478660, −4.31489484676743874612318866459, −3.82744557524133953554722235651, −3.56822262297053512783743584258, −2.69904648162715290918230431263, −2.41997882473594672394905060754, −2.16721683340337093038419781585, −1.27153017758547989564993590404, 0, 0, 1.27153017758547989564993590404, 2.16721683340337093038419781585, 2.41997882473594672394905060754, 2.69904648162715290918230431263, 3.56822262297053512783743584258, 3.82744557524133953554722235651, 4.31489484676743874612318866459, 5.05766962892158957405227478660, 5.28330711136303118673229284591, 5.36560366892489293594598388830, 6.20534235550408897063443770371, 6.25933822836416351247379396935, 6.98253808718326664738695357366, 7.18429845183093739347714495190, 7.86098078024792670901306255406, 8.031608597980090646472922657602, 8.175334856283479565184366458135, 8.758013295792964580189445896610

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