Properties

Label 4-4312-1.1-c1e2-0-3
Degree $4$
Conductor $4312$
Sign $1$
Analytic cond. $0.274936$
Root an. cond. $0.724116$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 11-s + 16-s − 6·17-s − 2·19-s − 2·20-s + 22-s − 4·23-s + 2·25-s + 2·31-s + 32-s − 6·34-s + 2·37-s − 2·38-s − 2·40-s + 2·41-s + 4·43-s + 44-s − 4·46-s − 2·47-s − 49-s + 2·50-s − 2·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 0.301·11-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s + 2/5·25-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.324·38-s − 0.316·40-s + 0.312·41-s + 0.609·43-s + 0.150·44-s − 0.589·46-s − 0.291·47-s − 1/7·49-s + 0.282·50-s − 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4312\)    =    \(2^{3} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.274936\)
Root analytic conductor: \(0.724116\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4312} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028259700\)
\(L(\frac12)\) \(\approx\) \(1.028259700\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 170 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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

−17.8369964058, −17.0434285046, −16.6183702981, −15.9909782834, −15.5390760474, −15.2491223779, −14.6074992840, −14.0382099450, −13.5828602765, −12.8172585107, −12.5505779282, −11.7405240279, −11.4430328108, −10.8364122626, −10.2519107770, −9.35689313101, −8.72782614033, −8.00759119881, −7.43974847057, −6.56734068869, −6.14777980938, −4.97164236309, −4.30225395490, −3.62488793906, −2.33149494969, 2.33149494969, 3.62488793906, 4.30225395490, 4.97164236309, 6.14777980938, 6.56734068869, 7.43974847057, 8.00759119881, 8.72782614033, 9.35689313101, 10.2519107770, 10.8364122626, 11.4430328108, 11.7405240279, 12.5505779282, 12.8172585107, 13.5828602765, 14.0382099450, 14.6074992840, 15.2491223779, 15.5390760474, 15.9909782834, 16.6183702981, 17.0434285046, 17.8369964058

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