Properties

Label 4-7600e2-1.1-c1e2-0-4
Degree $4$
Conductor $57760000$
Sign $1$
Analytic cond. $3682.82$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 6·7-s − 9-s − 6·13-s − 2·17-s + 2·19-s + 12·21-s + 10·23-s − 6·27-s − 6·29-s − 4·31-s − 12·39-s + 12·43-s + 15·49-s − 4·51-s + 6·53-s + 4·57-s + 6·59-s + 20·61-s − 6·63-s + 18·67-s + 20·69-s + 24·71-s − 6·73-s − 4·79-s − 4·81-s + 12·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 2.26·7-s − 1/3·9-s − 1.66·13-s − 0.485·17-s + 0.458·19-s + 2.61·21-s + 2.08·23-s − 1.15·27-s − 1.11·29-s − 0.718·31-s − 1.92·39-s + 1.82·43-s + 15/7·49-s − 0.560·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s + 2.56·61-s − 0.755·63-s + 2.19·67-s + 2.40·69-s + 2.84·71-s − 0.702·73-s − 0.450·79-s − 4/9·81-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57760000\)    =    \(2^{8} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3682.82\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57760000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.975143310\)
\(L(\frac12)\) \(\approx\) \(6.975143310\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 24 T + 4 p T^{2} - 24 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_4$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−7.976584807959091217271511858297, −7.82016010939966412617273180562, −7.36243594287606316923766247107, −7.28557645294643647943438977413, −6.85083730224638853489088562510, −6.48639967789367723116518139786, −5.61399051349855240940904118844, −5.57449215302970781917805708784, −5.08528312481028799144300779478, −5.00233746173306346607435648553, −4.60830474472460529232582775294, −4.10994176468701710379574596263, −3.56843083734713254058273416968, −3.46783207734431914887480932850, −2.62057138152629032964333986600, −2.48829895444736018691039011798, −2.06754149125574246855682069547, −1.89270961611857388709543818137, −0.932759353704666590548488436690, −0.67747242566603682892335784678, 0.67747242566603682892335784678, 0.932759353704666590548488436690, 1.89270961611857388709543818137, 2.06754149125574246855682069547, 2.48829895444736018691039011798, 2.62057138152629032964333986600, 3.46783207734431914887480932850, 3.56843083734713254058273416968, 4.10994176468701710379574596263, 4.60830474472460529232582775294, 5.00233746173306346607435648553, 5.08528312481028799144300779478, 5.57449215302970781917805708784, 5.61399051349855240940904118844, 6.48639967789367723116518139786, 6.85083730224638853489088562510, 7.28557645294643647943438977413, 7.36243594287606316923766247107, 7.82016010939966412617273180562, 7.976584807959091217271511858297

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