Properties

Label 4-7616e2-1.1-c1e2-0-9
Degree $4$
Conductor $58003456$
Sign $1$
Analytic cond. $3698.35$
Root an. cond. $7.79833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 2·7-s − 2·9-s − 6·13-s − 15-s + 2·17-s − 6·19-s − 2·21-s − 6·25-s + 2·27-s + 5·31-s + 2·35-s − 2·37-s + 6·39-s − 3·41-s − 11·43-s − 2·45-s + 2·47-s + 3·49-s − 2·51-s + 11·53-s + 6·57-s − 4·59-s + 5·61-s − 4·63-s − 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s − 1.37·19-s − 0.436·21-s − 6/5·25-s + 0.384·27-s + 0.898·31-s + 0.338·35-s − 0.328·37-s + 0.960·39-s − 0.468·41-s − 1.67·43-s − 0.298·45-s + 0.291·47-s + 3/7·49-s − 0.280·51-s + 1.51·53-s + 0.794·57-s − 0.520·59-s + 0.640·61-s − 0.503·63-s − 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(58003456\)    =    \(2^{12} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(3698.35\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 58003456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 5 T + 65 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 11 T + 113 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 143 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 26 T + 334 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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

−7.63016014022101103487022071276, −7.38988856810408992250970188395, −6.88504614651688014263683805649, −6.77770314294095575671931760564, −6.10401489682865978310945972154, −6.09004899815329670739141928375, −5.44876031132231184750358307196, −5.34568633083706373941449836479, −4.92131998321942959458128814767, −4.71567236984094831812065547465, −4.03485968359762018094956987200, −3.95725226192777045142416743743, −3.32315965565525780387708938368, −2.67231087050597624240718026027, −2.48137675798750353652743958826, −2.06699032090843410407735925647, −1.59414313682064723442830397276, −1.00196639664323307928118727823, 0, 0, 1.00196639664323307928118727823, 1.59414313682064723442830397276, 2.06699032090843410407735925647, 2.48137675798750353652743958826, 2.67231087050597624240718026027, 3.32315965565525780387708938368, 3.95725226192777045142416743743, 4.03485968359762018094956987200, 4.71567236984094831812065547465, 4.92131998321942959458128814767, 5.34568633083706373941449836479, 5.44876031132231184750358307196, 6.09004899815329670739141928375, 6.10401489682865978310945972154, 6.77770314294095575671931760564, 6.88504614651688014263683805649, 7.38988856810408992250970188395, 7.63016014022101103487022071276

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