Properties

Label 4-600e2-1.1-c3e2-0-11
Degree $4$
Conductor $360000$
Sign $1$
Analytic cond. $1253.24$
Root an. cond. $5.94988$
Motivic weight $3$
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  + 6·3-s + 6·7-s + 27·9-s + 8·11-s + 14·13-s + 86·19-s + 36·21-s + 128·23-s + 108·27-s + 344·29-s − 158·31-s + 48·33-s + 36·37-s + 84·39-s − 244·41-s + 390·43-s + 756·47-s + 65·49-s + 268·53-s + 516·57-s + 4·59-s − 1.03e3·61-s + 162·63-s + 1.62e3·67-s + 768·69-s − 276·71-s + 644·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.323·7-s + 9-s + 0.219·11-s + 0.298·13-s + 1.03·19-s + 0.374·21-s + 1.16·23-s + 0.769·27-s + 2.20·29-s − 0.915·31-s + 0.253·33-s + 0.159·37-s + 0.344·39-s − 0.929·41-s + 1.38·43-s + 2.34·47-s + 0.189·49-s + 0.694·53-s + 1.19·57-s + 0.00882·59-s − 2.17·61-s + 0.323·63-s + 2.95·67-s + 1.33·69-s − 0.461·71-s + 1.03·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1253.24\)
Root analytic conductor: \(5.94988\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 360000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.830079004\)
\(L(\frac12)\) \(\approx\) \(6.830079004\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 6 T - 29 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 8 T + 1954 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 14 T + 119 p T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9102 T^{2} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 86 T + 9051 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 128 T + 21914 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 344 T + 2078 p T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 158 T + 30347 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 36 T + 75566 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 244 T + 117250 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 390 T + 161563 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 756 T + 338946 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 268 T + 280234 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 364426 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1034 T + 674915 T^{2} + 1034 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1622 T + 1200603 T^{2} - 1622 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 276 T - 53570 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 644 T + 809318 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 944 T + 919262 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 484 T + 460762 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 2224 T + 2634898 T^{2} - 2224 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 510 T + 1148995 T^{2} - 510 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.22424895040710987877594552685, −10.21781173763787927704138171377, −9.369287988429713426074232524048, −9.181609096025459352742041025068, −8.627996714885208179457385583371, −8.544679324482166286350023546733, −7.72918234515265951043783930536, −7.50347106197718052600376477478, −7.07177229955207041599217979624, −6.55077329554555578063795553575, −5.94577138488621769172773437157, −5.39617432774299814727200026802, −4.73108867840722971064069825243, −4.45787768119235818571513673796, −3.55292561400285019641442499223, −3.39795607996282883455017078929, −2.56544465145209007411140287991, −2.20486122751630337498074028331, −1.15178239057919236019958903402, −0.853404867179953651216001202268, 0.853404867179953651216001202268, 1.15178239057919236019958903402, 2.20486122751630337498074028331, 2.56544465145209007411140287991, 3.39795607996282883455017078929, 3.55292561400285019641442499223, 4.45787768119235818571513673796, 4.73108867840722971064069825243, 5.39617432774299814727200026802, 5.94577138488621769172773437157, 6.55077329554555578063795553575, 7.07177229955207041599217979624, 7.50347106197718052600376477478, 7.72918234515265951043783930536, 8.544679324482166286350023546733, 8.627996714885208179457385583371, 9.181609096025459352742041025068, 9.369287988429713426074232524048, 10.21781173763787927704138171377, 10.22424895040710987877594552685

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