Properties

Label 4-1012-1.1-c1e2-0-0
Degree $4$
Conductor $1012$
Sign $1$
Analytic cond. $0.0645259$
Root an. cond. $0.504003$
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  − 3-s + 4-s − 4·5-s + 7-s − 2·11-s − 12-s − 13-s + 4·15-s + 16-s + 17-s − 19-s − 4·20-s − 21-s + 8·23-s + 6·25-s − 2·27-s + 28-s + 2·29-s + 3·31-s + 2·33-s − 4·35-s − 2·37-s + 39-s − 3·41-s + 7·43-s − 2·44-s − 47-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.377·7-s − 0.603·11-s − 0.288·12-s − 0.277·13-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.229·19-s − 0.894·20-s − 0.218·21-s + 1.66·23-s + 6/5·25-s − 0.384·27-s + 0.188·28-s + 0.371·29-s + 0.538·31-s + 0.348·33-s − 0.676·35-s − 0.328·37-s + 0.160·39-s − 0.468·41-s + 1.06·43-s − 0.301·44-s − 0.145·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.0645259\)
Root analytic conductor: \(0.504003\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1012} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1012,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4115326478\)
\(L(\frac12)\) \(\approx\) \(0.4115326478\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 9 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - T - p T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T - 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 73 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 120 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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

−19.9882803913, −19.2563394268, −19.0571681562, −18.5213982025, −17.5206958166, −17.2281843436, −16.5311236336, −15.8430383489, −15.6040108100, −14.9999989225, −14.4792364258, −13.4887593385, −12.6909368687, −12.1161852294, −11.6782886947, −10.9380100660, −10.7839026062, −9.61539365714, −8.56784628748, −7.84388923320, −7.38627983724, −6.48476905981, −5.27481627094, −4.42148892137, −3.15322481656, 3.15322481656, 4.42148892137, 5.27481627094, 6.48476905981, 7.38627983724, 7.84388923320, 8.56784628748, 9.61539365714, 10.7839026062, 10.9380100660, 11.6782886947, 12.1161852294, 12.6909368687, 13.4887593385, 14.4792364258, 14.9999989225, 15.6040108100, 15.8430383489, 16.5311236336, 17.2281843436, 17.5206958166, 18.5213982025, 19.0571681562, 19.2563394268, 19.9882803913

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