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Properties

Label 4-259200-1.1-c1e2-0-28
Degree $4$
Conductor $259200$
Sign $1$
Analytic cond. $16.5268$
Root an. cond. $2.01626$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 16-s − 2·19-s + 4·20-s − 2·23-s + 11·25-s + 4·29-s − 32-s + 2·38-s − 4·40-s + 4·43-s + 2·46-s + 12·47-s + 4·49-s − 11·50-s + 16·53-s − 4·58-s + 64-s − 10·67-s − 2·71-s − 4·73-s − 2·76-s + 4·80-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1/4·16-s − 0.458·19-s + 0.894·20-s − 0.417·23-s + 11/5·25-s + 0.742·29-s − 0.176·32-s + 0.324·38-s − 0.632·40-s + 0.609·43-s + 0.294·46-s + 1.75·47-s + 4/7·49-s − 1.55·50-s + 2.19·53-s − 0.525·58-s + 1/8·64-s − 1.22·67-s − 0.237·71-s − 0.468·73-s − 0.229·76-s + 0.447·80-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 259200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842117682\)
\(L(\frac12)\) \(\approx\) \(1.842117682\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.c_o
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.c_w
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ae_bu
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.am_ew
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.aq_gk
59$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.59.a_e
61$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.61.a_abm
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.k_fe
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.c_dq
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.e_fe
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.83.a_g
89$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.89.a_acw
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.k_go
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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

−9.081444259937376947085319925841, −8.523285408969769437469971320630, −8.228116244396256443869053669012, −7.34470332136653973429739824837, −7.11957178074071411009733756910, −6.54351622473128791008628510863, −5.94692085854568052038423647429, −5.77270774455158360582234041973, −5.22074566980770104380699023481, −4.46117196245196832818359739611, −3.88296031384893801501633766656, −2.81751947676589648650979183563, −2.48071790774233938628729689497, −1.79620548558312040514035893609, −0.961937939221127316723838522284, 0.961937939221127316723838522284, 1.79620548558312040514035893609, 2.48071790774233938628729689497, 2.81751947676589648650979183563, 3.88296031384893801501633766656, 4.46117196245196832818359739611, 5.22074566980770104380699023481, 5.77270774455158360582234041973, 5.94692085854568052038423647429, 6.54351622473128791008628510863, 7.11957178074071411009733756910, 7.34470332136653973429739824837, 8.228116244396256443869053669012, 8.523285408969769437469971320630, 9.081444259937376947085319925841

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