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Properties

Label 4-259200-1.1-c1e2-0-72
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 12·11-s + 16-s + 12·17-s − 8·19-s − 12·22-s + 25-s + 32-s + 12·34-s − 8·38-s + 16·43-s − 12·44-s − 10·49-s + 50-s − 12·59-s + 64-s − 8·67-s + 12·68-s − 20·73-s − 8·76-s − 24·83-s + 16·86-s − 12·88-s − 24·89-s + 4·97-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 3.61·11-s + 1/4·16-s + 2.91·17-s − 1.83·19-s − 2.55·22-s + 1/5·25-s + 0.176·32-s + 2.05·34-s − 1.29·38-s + 2.43·43-s − 1.80·44-s − 1.42·49-s + 0.141·50-s − 1.56·59-s + 1/8·64-s − 0.977·67-s + 1.45·68-s − 2.34·73-s − 0.917·76-s − 2.63·83-s + 1.72·86-s − 1.27·88-s − 2.54·89-s + 0.406·97-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$C_1$ \( 1 - T \)
3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
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_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
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

−8.330531568433955397282390142839, −8.137806933221736892131373840887, −7.73490121522356695217356839933, −7.42257774366004622063287945506, −6.86802385357238762255411937862, −5.84293355103774809066040280829, −5.62305662493911647485930324213, −5.57449936126795434832243724178, −4.56211949870166416184694202500, −4.48715934481021693344557314737, −3.37793372322877666574139691343, −2.75202257331423339947272598173, −2.68907161640043859094582658785, −1.53717311332283821140901478063, 0, 1.53717311332283821140901478063, 2.68907161640043859094582658785, 2.75202257331423339947272598173, 3.37793372322877666574139691343, 4.48715934481021693344557314737, 4.56211949870166416184694202500, 5.57449936126795434832243724178, 5.62305662493911647485930324213, 5.84293355103774809066040280829, 6.86802385357238762255411937862, 7.42257774366004622063287945506, 7.73490121522356695217356839933, 8.137806933221736892131373840887, 8.330531568433955397282390142839

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