Properties

Label 4-784000-1.1-c1e2-0-4
Degree $4$
Conductor $784000$
Sign $1$
Analytic cond. $49.9885$
Root an. cond. $2.65899$
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-s + 4-s + 5-s − 8-s − 6·9-s − 10-s + 12·13-s + 16-s + 6·18-s + 20-s + 25-s − 12·26-s + 16·31-s − 32-s − 6·36-s + 20·37-s − 40-s + 4·41-s − 8·43-s − 6·45-s + 49-s − 50-s + 12·52-s + 4·53-s − 16·62-s + 64-s + 12·65-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 2·9-s − 0.316·10-s + 3.32·13-s + 1/4·16-s + 1.41·18-s + 0.223·20-s + 1/5·25-s − 2.35·26-s + 2.87·31-s − 0.176·32-s − 36-s + 3.28·37-s − 0.158·40-s + 0.624·41-s − 1.21·43-s − 0.894·45-s + 1/7·49-s − 0.141·50-s + 1.66·52-s + 0.549·53-s − 2.03·62-s + 1/8·64-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(784000\)    =    \(2^{7} \cdot 5^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(49.9885\)
Root analytic conductor: \(2.65899\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 784000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.762170815\)
\(L(\frac12)\) \(\approx\) \(1.762170815\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.342925504546681427970604924895, −8.089729535357517933440155073370, −7.63810043151096799752774274091, −6.56718920518778472538358027386, −6.50661147582867110066747443111, −6.01430128602422585899125610915, −5.82344169953684002039972565231, −5.35297783463712406492518278139, −4.36823223624486251008730105784, −4.07505442753619333270812979660, −3.11732069850814337090071774581, −2.98711940319117381089884417433, −2.34423344984301921891298897662, −1.27008084899723889248060652436, −0.837608542604674414323310984226, 0.837608542604674414323310984226, 1.27008084899723889248060652436, 2.34423344984301921891298897662, 2.98711940319117381089884417433, 3.11732069850814337090071774581, 4.07505442753619333270812979660, 4.36823223624486251008730105784, 5.35297783463712406492518278139, 5.82344169953684002039972565231, 6.01430128602422585899125610915, 6.50661147582867110066747443111, 6.56718920518778472538358027386, 7.63810043151096799752774274091, 8.089729535357517933440155073370, 8.342925504546681427970604924895

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