Properties

Label 2-2142-1.1-c1-0-9
Degree $2$
Conductor $2142$
Sign $1$
Analytic cond. $17.1039$
Root an. cond. $4.13569$
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 − 7-s − 8-s − 2·10-s − 6·13-s + 14-s + 16-s − 17-s + 2·20-s + 8·23-s − 25-s + 6·26-s − 28-s + 6·29-s − 8·31-s − 32-s + 34-s − 2·35-s + 10·37-s − 2·40-s + 6·41-s + 12·43-s − 8·46-s + 49-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.447·20-s + 1.66·23-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.171·34-s − 0.338·35-s + 1.64·37-s − 0.316·40-s + 0.937·41-s + 1.82·43-s − 1.17·46-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(17.1039\)
Root analytic conductor: \(4.13569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2142} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2142,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306032340\)
\(L(\frac12)\) \(\approx\) \(1.306032340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.352367073730369250319535139199, −8.466864404810002345434086097975, −7.34947934570743894088685704902, −7.02175028110277092253993140442, −5.96218325167265543606565738451, −5.29944243683183488334817629056, −4.24467666749940526398113891006, −2.78295861731664811401382989703, −2.27199331297833251963893517833, −0.824579001110863552316391184137, 0.824579001110863552316391184137, 2.27199331297833251963893517833, 2.78295861731664811401382989703, 4.24467666749940526398113891006, 5.29944243683183488334817629056, 5.96218325167265543606565738451, 7.02175028110277092253993140442, 7.34947934570743894088685704902, 8.466864404810002345434086097975, 9.352367073730369250319535139199

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