Properties

Label 4-40e4-1.1-c1e2-0-35
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  + 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

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

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: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.551617374\)
\(L(\frac12)\) \(\approx\) \(5.551617374\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−9.363258489822003334166133824785, −8.949308776173010957975980837711, −8.615438685139658252791054915969, −8.454380564982761455140661602334, −8.217952852428854810315473712050, −7.925309679445865231984154554998, −7.11841544533514631319757746441, −7.07947089281660569271320211210, −6.31477005881578345439531718352, −6.11097635043789657226929159402, −5.35193605369363449701521096422, −4.94092230917738424743200509192, −4.57260612362092408740704908740, −4.18423420983815244687014448918, −3.58089537743508553096294215741, −3.30927135895031105960132761393, −2.24730671604403008144562798173, −2.17910011568493508616522677277, −1.56039085313696789733358049108, −0.911043372697545569805261962514, 0.911043372697545569805261962514, 1.56039085313696789733358049108, 2.17910011568493508616522677277, 2.24730671604403008144562798173, 3.30927135895031105960132761393, 3.58089537743508553096294215741, 4.18423420983815244687014448918, 4.57260612362092408740704908740, 4.94092230917738424743200509192, 5.35193605369363449701521096422, 6.11097635043789657226929159402, 6.31477005881578345439531718352, 7.07947089281660569271320211210, 7.11841544533514631319757746441, 7.925309679445865231984154554998, 8.217952852428854810315473712050, 8.454380564982761455140661602334, 8.615438685139658252791054915969, 8.949308776173010957975980837711, 9.363258489822003334166133824785

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