Properties

Label 4-7800e2-1.1-c1e2-0-4
Degree $4$
Conductor $60840000$
Sign $1$
Analytic cond. $3879.21$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 3·9-s − 2·11-s + 2·13-s − 2·17-s − 4·19-s − 4·21-s − 4·23-s + 4·27-s − 6·29-s + 6·31-s − 4·33-s + 4·37-s + 4·39-s + 4·43-s − 10·47-s − 9·49-s − 4·51-s + 6·53-s − 8·57-s − 10·59-s + 2·61-s − 6·63-s − 10·67-s − 8·69-s − 4·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 9-s − 0.603·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 0.769·27-s − 1.11·29-s + 1.07·31-s − 0.696·33-s + 0.657·37-s + 0.640·39-s + 0.609·43-s − 1.45·47-s − 9/7·49-s − 0.560·51-s + 0.824·53-s − 1.05·57-s − 1.30·59-s + 0.256·61-s − 0.755·63-s − 1.22·67-s − 0.963·69-s − 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60840000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3879.21\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 60840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 69 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 141 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 115 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 141 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_4$ \( 1 - 4 T - 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.85169013255085771952788934668, −7.49662425583364825893154870058, −6.86861963198206200584581146741, −6.74447949523242070017567938095, −6.29375670655904299077238619417, −6.15197511404963312029320639393, −5.59761566725366290860496508345, −5.25548477135529600953360069665, −4.75281748996854592338607619198, −4.28478198181978196413609115378, −4.08813232584514651253703421161, −3.76168783065216142658572618837, −3.16346027183524709894427240810, −2.95835306723105811582039678931, −2.46410789823083094832853069126, −2.23760123399759453338661676653, −1.45145032964540209968022734204, −1.32667992047723484410723322672, 0, 0, 1.32667992047723484410723322672, 1.45145032964540209968022734204, 2.23760123399759453338661676653, 2.46410789823083094832853069126, 2.95835306723105811582039678931, 3.16346027183524709894427240810, 3.76168783065216142658572618837, 4.08813232584514651253703421161, 4.28478198181978196413609115378, 4.75281748996854592338607619198, 5.25548477135529600953360069665, 5.59761566725366290860496508345, 6.15197511404963312029320639393, 6.29375670655904299077238619417, 6.74447949523242070017567938095, 6.86861963198206200584581146741, 7.49662425583364825893154870058, 7.85169013255085771952788934668

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