Properties

Label 4-6720e2-1.1-c1e2-0-11
Degree $4$
Conductor $45158400$
Sign $1$
Analytic cond. $2879.33$
Root an. cond. $7.32526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 2·7-s + 3·9-s + 4·11-s + 4·15-s + 4·17-s + 4·19-s + 4·21-s + 3·25-s + 4·27-s + 4·29-s + 4·31-s + 8·33-s + 4·35-s − 4·37-s + 4·41-s + 8·43-s + 6·45-s − 8·47-s + 3·49-s + 8·51-s + 8·53-s + 8·55-s + 8·57-s − 4·61-s + 6·63-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.20·11-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.872·21-s + 3/5·25-s + 0.769·27-s + 0.742·29-s + 0.718·31-s + 1.39·33-s + 0.676·35-s − 0.657·37-s + 0.624·41-s + 1.21·43-s + 0.894·45-s − 1.16·47-s + 3/7·49-s + 1.12·51-s + 1.09·53-s + 1.07·55-s + 1.05·57-s − 0.512·61-s + 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(45158400\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2879.33\)
Root analytic conductor: \(7.32526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 45158400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.04785760\)
\(L(\frac12)\) \(\approx\) \(11.04785760\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 16 T + 182 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 32 T + 438 T^{2} - 32 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.010227223424037777023276346578, −7.960477218782935313877047061817, −7.47258883289462418995625075196, −7.24670320467552340285557954342, −6.73795803524483661540150029468, −6.52699494972866303408195148801, −5.91465731724574741411148777371, −5.86901074409228539545046329765, −5.22763322908266045715430049415, −4.96648975671923527152762028974, −4.50755989249754518876189545895, −4.20193786581538460053364416978, −3.67119742975943750017667206708, −3.38207727875111326199671576545, −2.88375344198569343885894375532, −2.60102662042742863051712115322, −1.95465073649489617839853472416, −1.70232720376224871737973823393, −1.05660785103639911918143859502, −0.899532293161421978313285815201, 0.899532293161421978313285815201, 1.05660785103639911918143859502, 1.70232720376224871737973823393, 1.95465073649489617839853472416, 2.60102662042742863051712115322, 2.88375344198569343885894375532, 3.38207727875111326199671576545, 3.67119742975943750017667206708, 4.20193786581538460053364416978, 4.50755989249754518876189545895, 4.96648975671923527152762028974, 5.22763322908266045715430049415, 5.86901074409228539545046329765, 5.91465731724574741411148777371, 6.52699494972866303408195148801, 6.73795803524483661540150029468, 7.24670320467552340285557954342, 7.47258883289462418995625075196, 7.960477218782935313877047061817, 8.010227223424037777023276346578

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