Properties

Label 4-40e4-1.1-c1e2-0-23
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4·7-s + 8·9-s − 2·13-s + 10·17-s + 8·19-s − 16·21-s + 4·23-s − 12·27-s + 2·37-s + 8·39-s + 12·43-s − 4·47-s + 8·49-s − 40·51-s − 14·53-s − 32·57-s + 8·59-s + 8·61-s + 32·63-s + 20·67-s − 16·69-s + 6·73-s − 32·79-s + 23·81-s + 4·83-s − 8·91-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.51·7-s + 8/3·9-s − 0.554·13-s + 2.42·17-s + 1.83·19-s − 3.49·21-s + 0.834·23-s − 2.30·27-s + 0.328·37-s + 1.28·39-s + 1.82·43-s − 0.583·47-s + 8/7·49-s − 5.60·51-s − 1.92·53-s − 4.23·57-s + 1.04·59-s + 1.02·61-s + 4.03·63-s + 2.44·67-s − 1.92·69-s + 0.702·73-s − 3.60·79-s + 23/9·81-s + 0.439·83-s − 0.838·91-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

\(L(1)\) \(\approx\) \(1.529943574\)
\(L(\frac12)\) \(\approx\) \(1.529943574\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_i
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ak_by
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_i
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_c
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_cu
47$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_i
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.53.o_du
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.67.au_hs
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_s
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_i
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_s
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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

−9.824035746050276995318001059359, −9.413622223939550624798789734255, −8.879497359957271322294128083006, −8.133125465690917920606161970208, −7.910183342301211870629676942020, −7.61727826073966844802561949689, −7.00621352936811647541120539422, −6.99354812874300665672723799699, −6.10907772588770603036871078024, −5.76375537033056747436714548833, −5.37475069642031071734892128175, −5.37128852542097409721740056601, −4.75438873219656222026554312231, −4.58476895883111918494663699182, −3.71702910636198709638027100023, −3.29613539383094573066654059563, −2.54172334288598827486348885621, −1.64414702522611555776394123864, −1.04196385229980683940419055957, −0.74224336133570552787174303818, 0.74224336133570552787174303818, 1.04196385229980683940419055957, 1.64414702522611555776394123864, 2.54172334288598827486348885621, 3.29613539383094573066654059563, 3.71702910636198709638027100023, 4.58476895883111918494663699182, 4.75438873219656222026554312231, 5.37128852542097409721740056601, 5.37475069642031071734892128175, 5.76375537033056747436714548833, 6.10907772588770603036871078024, 6.99354812874300665672723799699, 7.00621352936811647541120539422, 7.61727826073966844802561949689, 7.910183342301211870629676942020, 8.133125465690917920606161970208, 8.879497359957271322294128083006, 9.413622223939550624798789734255, 9.824035746050276995318001059359

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