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Properties

Label 4-259200-1.1-c1e2-0-75
Degree $4$
Conductor $259200$
Sign $-1$
Analytic cond. $16.5268$
Root an. cond. $2.01626$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 12·13-s + 4·17-s + 3·25-s + 12·29-s − 12·37-s − 20·41-s + 2·49-s − 20·53-s + 12·61-s − 24·65-s − 28·73-s + 8·85-s − 4·89-s + 4·97-s + 28·101-s − 20·109-s − 12·113-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.894·5-s − 3.32·13-s + 0.970·17-s + 3/5·25-s + 2.22·29-s − 1.97·37-s − 3.12·41-s + 2/7·49-s − 2.74·53-s + 1.53·61-s − 2.97·65-s − 3.27·73-s + 0.867·85-s − 0.423·89-s + 0.406·97-s + 2.78·101-s − 1.91·109-s − 1.12·113-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(259200\)    =    \(2^{7} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(16.5268\)
Root analytic conductor: \(2.01626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 259200,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_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

−8.609245026512276947697146238421, −8.312681871958482675893970014801, −7.67822247307498044046677545125, −7.18880346144848821197627995561, −6.82423053307536595497404589251, −6.46041242786302870623045801913, −5.61049576810701540702833296151, −5.12597312940943561354086695880, −4.92167313563440441781925895939, −4.43140184316888462520939708171, −3.20808194979498891977471116338, −2.97865181251010540219662645668, −2.19282779410354418054159546466, −1.55129913823425505871935815725, 0, 1.55129913823425505871935815725, 2.19282779410354418054159546466, 2.97865181251010540219662645668, 3.20808194979498891977471116338, 4.43140184316888462520939708171, 4.92167313563440441781925895939, 5.12597312940943561354086695880, 5.61049576810701540702833296151, 6.46041242786302870623045801913, 6.82423053307536595497404589251, 7.18880346144848821197627995561, 7.67822247307498044046677545125, 8.312681871958482675893970014801, 8.609245026512276947697146238421

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