Properties

Label 4-40e4-1.1-c1e2-0-21
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  − 2·3-s + 6·7-s + 2·9-s − 12·21-s + 2·23-s − 6·27-s + 12·29-s − 18·43-s + 14·47-s + 18·49-s + 12·63-s − 6·67-s − 4·69-s + 11·81-s + 22·83-s − 24·87-s + 12·89-s + 36·101-s + 18·103-s + 26·107-s − 22·121-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.26·7-s + 2/3·9-s − 2.61·21-s + 0.417·23-s − 1.15·27-s + 2.22·29-s − 2.74·43-s + 2.04·47-s + 18/7·49-s + 1.51·63-s − 0.733·67-s − 0.481·69-s + 11/9·81-s + 2.41·83-s − 2.57·87-s + 1.27·89-s + 3.58·101-s + 1.77·103-s + 2.51·107-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.159837556\)
\(L(\frac12)\) \(\approx\) \(2.159837556\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_s
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_c
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.43.s_gg
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_du
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_s
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.83.aw_ji
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−9.808634071172554708725211779728, −8.981021072051790332711909896389, −8.715682100582172526057577956387, −8.553190499172138405910183511041, −7.86922279576880230218534033104, −7.62561266042530916678711360244, −7.35729692236798593523829620330, −6.72016697862793868423748699054, −6.17364264952357770932618056764, −6.10430167146131886553183041943, −5.29369115774265646522695955691, −5.06187783798438436590623467379, −4.69630400834468146692411967977, −4.55449588525619354671206126566, −3.72895650149875186242656031563, −3.27386988931851935633473820789, −2.31422058439203483863311661110, −1.96314886680260062258064664354, −1.24744484253273168861608210071, −0.69687543080154082339848196756, 0.69687543080154082339848196756, 1.24744484253273168861608210071, 1.96314886680260062258064664354, 2.31422058439203483863311661110, 3.27386988931851935633473820789, 3.72895650149875186242656031563, 4.55449588525619354671206126566, 4.69630400834468146692411967977, 5.06187783798438436590623467379, 5.29369115774265646522695955691, 6.10430167146131886553183041943, 6.17364264952357770932618056764, 6.72016697862793868423748699054, 7.35729692236798593523829620330, 7.62561266042530916678711360244, 7.86922279576880230218534033104, 8.553190499172138405910183511041, 8.715682100582172526057577956387, 8.981021072051790332711909896389, 9.808634071172554708725211779728

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