Properties

Label 4-3040e2-1.1-c1e2-0-0
Degree $4$
Conductor $9241600$
Sign $1$
Analytic cond. $589.252$
Root an. cond. $4.92691$
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  + 3-s − 2·5-s − 5·7-s − 9-s + 8·11-s + 13-s − 2·15-s + 3·17-s + 2·19-s − 5·21-s − 23-s + 3·25-s − 11·29-s + 2·31-s + 8·33-s + 10·35-s + 6·37-s + 39-s + 4·43-s + 2·45-s + 2·47-s + 9·49-s + 3·51-s + 9·53-s − 16·55-s + 2·57-s − 5·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.88·7-s − 1/3·9-s + 2.41·11-s + 0.277·13-s − 0.516·15-s + 0.727·17-s + 0.458·19-s − 1.09·21-s − 0.208·23-s + 3/5·25-s − 2.04·29-s + 0.359·31-s + 1.39·33-s + 1.69·35-s + 0.986·37-s + 0.160·39-s + 0.609·43-s + 0.298·45-s + 0.291·47-s + 9/7·49-s + 0.420·51-s + 1.23·53-s − 2.15·55-s + 0.264·57-s − 0.650·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9241600\)    =    \(2^{10} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(589.252\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9241600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.512772472\)
\(L(\frac12)\) \(\approx\) \(2.512772472\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 9 T + 122 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 18 T + 186 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 17 T + 202 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.895318358094823467828782393939, −8.738740134275015222602293400618, −8.094466549504147361129515408084, −7.85381428248481339348840369631, −7.22262196112032373566630806190, −7.19151073744191981336999668427, −6.63916936997000433512436559977, −6.28717075139434647357532824837, −5.91786885671210340845076591957, −5.78688518457838820853449942160, −4.94046561754160211624714569205, −4.46522482451246616473928113876, −3.99635257585489615916870733103, −3.54985649169236479077483425639, −3.34793456630333448937126847499, −3.29266600131638583290883712182, −2.35582090438890933964225614140, −1.90472763444663391585263956543, −0.948917446234411268788742012051, −0.61268327994003756232515891643, 0.61268327994003756232515891643, 0.948917446234411268788742012051, 1.90472763444663391585263956543, 2.35582090438890933964225614140, 3.29266600131638583290883712182, 3.34793456630333448937126847499, 3.54985649169236479077483425639, 3.99635257585489615916870733103, 4.46522482451246616473928113876, 4.94046561754160211624714569205, 5.78688518457838820853449942160, 5.91786885671210340845076591957, 6.28717075139434647357532824837, 6.63916936997000433512436559977, 7.19151073744191981336999668427, 7.22262196112032373566630806190, 7.85381428248481339348840369631, 8.094466549504147361129515408084, 8.738740134275015222602293400618, 8.895318358094823467828782393939

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