Properties

Label 4-7728e2-1.1-c1e2-0-2
Degree $4$
Conductor $59721984$
Sign $1$
Analytic cond. $3807.92$
Root an. cond. $7.85546$
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 + 5-s − 2·7-s + 3·9-s − 11-s + 5·13-s + 2·15-s + 8·17-s − 9·19-s − 4·21-s − 2·23-s − 25-s + 4·27-s + 5·29-s + 12·31-s − 2·33-s − 2·35-s − 11·37-s + 10·39-s − 43-s + 3·45-s + 4·47-s + 3·49-s + 16·51-s − 3·53-s − 55-s − 18·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 0.755·7-s + 9-s − 0.301·11-s + 1.38·13-s + 0.516·15-s + 1.94·17-s − 2.06·19-s − 0.872·21-s − 0.417·23-s − 1/5·25-s + 0.769·27-s + 0.928·29-s + 2.15·31-s − 0.348·33-s − 0.338·35-s − 1.80·37-s + 1.60·39-s − 0.152·43-s + 0.447·45-s + 0.583·47-s + 3/7·49-s + 2.24·51-s − 0.412·53-s − 0.134·55-s − 2.38·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(59721984\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3807.92\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 59721984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.089382376\)
\(L(\frac12)\) \(\approx\) \(5.089382376\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T + 78 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 65 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 13 T + 152 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 144 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 14 T + 158 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−8.202231199743822179025670873930, −7.80406127815212048084138515442, −7.48600622705977314119406057216, −6.89695943845238943792126600277, −6.54007521929491323703797781594, −6.47347840048980373937922558827, −5.91560903418115008644298866922, −5.85344377678486701385229915042, −5.19153768102319321297844913269, −4.85399504610240050414086337419, −4.23007640608338524492251484060, −4.18735749690221273473664881667, −3.48541038078183103078959796312, −3.40249864551030348254460496356, −2.81754412317450386034233329442, −2.71196180262553176740451833899, −1.93928301331908762596962685366, −1.69254687299147460255472653897, −1.12398490094562238944069872369, −0.50953969450420984073089415385, 0.50953969450420984073089415385, 1.12398490094562238944069872369, 1.69254687299147460255472653897, 1.93928301331908762596962685366, 2.71196180262553176740451833899, 2.81754412317450386034233329442, 3.40249864551030348254460496356, 3.48541038078183103078959796312, 4.18735749690221273473664881667, 4.23007640608338524492251484060, 4.85399504610240050414086337419, 5.19153768102319321297844913269, 5.85344377678486701385229915042, 5.91560903418115008644298866922, 6.47347840048980373937922558827, 6.54007521929491323703797781594, 6.89695943845238943792126600277, 7.48600622705977314119406057216, 7.80406127815212048084138515442, 8.202231199743822179025670873930

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