Properties

Label 4-800e2-1.1-c1e2-0-7
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $40.8069$
Root an. cond. $2.52745$
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 − 6·13-s − 2·17-s + 8·19-s − 12·21-s + 2·23-s − 6·27-s + 2·37-s + 12·39-s − 20·41-s − 10·43-s + 6·47-s + 18·49-s + 4·51-s + 10·53-s − 16·57-s − 24·59-s + 4·61-s + 12·63-s + 2·67-s − 4·69-s − 2·73-s + 16·79-s + 11·81-s − 10·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.26·7-s + 2/3·9-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 2.61·21-s + 0.417·23-s − 1.15·27-s + 0.328·37-s + 1.92·39-s − 3.12·41-s − 1.52·43-s + 0.875·47-s + 18/7·49-s + 0.560·51-s + 1.37·53-s − 2.11·57-s − 3.12·59-s + 0.512·61-s + 1.51·63-s + 0.244·67-s − 0.481·69-s − 0.234·73-s + 1.80·79-s + 11/9·81-s − 1.09·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(40.8069\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387904343\)
\(L(\frac12)\) \(\approx\) \(1.387904343\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−10.53801898130760231602550792017, −10.04601612046077669039541069461, −9.881633541073524618857510399796, −9.177523187138729099815204687783, −8.845966807037717208827539248878, −8.055300510133531601149933734628, −8.043369381561794008056577988471, −7.29577776853698128231862495224, −7.20776009761738679295576243310, −6.68522468942190639094060403261, −5.84321312207294119439341714655, −5.51986834619945442717582828569, −5.04279350927061140336413948223, −4.70724086947760161543118487461, −4.63355241801052051179115717180, −3.58560007532189503648850603494, −2.98583320373619828613211231407, −1.86629275310895654054021106238, −1.78704947084357478927538519404, −0.64506924347125637965437535748, 0.64506924347125637965437535748, 1.78704947084357478927538519404, 1.86629275310895654054021106238, 2.98583320373619828613211231407, 3.58560007532189503648850603494, 4.63355241801052051179115717180, 4.70724086947760161543118487461, 5.04279350927061140336413948223, 5.51986834619945442717582828569, 5.84321312207294119439341714655, 6.68522468942190639094060403261, 7.20776009761738679295576243310, 7.29577776853698128231862495224, 8.043369381561794008056577988471, 8.055300510133531601149933734628, 8.845966807037717208827539248878, 9.177523187138729099815204687783, 9.881633541073524618857510399796, 10.04601612046077669039541069461, 10.53801898130760231602550792017

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