Properties

Label 2-161700-1.1-c1-0-5
Degree $2$
Conductor $161700$
Sign $1$
Analytic cond. $1291.18$
Root an. cond. $35.9330$
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 + 9-s + 11-s − 13-s − 5·19-s − 2·23-s − 27-s − 29-s − 8·31-s − 33-s − 37-s + 39-s − 6·43-s + 47-s + 2·53-s + 5·57-s − 9·59-s − 10·61-s − 7·67-s + 2·69-s + 9·73-s + 81-s + 87-s + 6·89-s + 8·93-s − 2·97-s + 99-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 1.14·19-s − 0.417·23-s − 0.192·27-s − 0.185·29-s − 1.43·31-s − 0.174·33-s − 0.164·37-s + 0.160·39-s − 0.914·43-s + 0.145·47-s + 0.274·53-s + 0.662·57-s − 1.17·59-s − 1.28·61-s − 0.855·67-s + 0.240·69-s + 1.05·73-s + 1/9·81-s + 0.107·87-s + 0.635·89-s + 0.829·93-s − 0.203·97-s + 0.100·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161700\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1291.18\)
Root analytic conductor: \(35.9330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{161700} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 161700,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4803997745\)
\(L(\frac12)\) \(\approx\) \(0.4803997745\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.23813123508366, −12.65071124775741, −12.34449547499763, −11.92283081691368, −11.27834939642221, −10.95094167111918, −10.48031087921827, −10.00406172572555, −9.463527205873762, −8.947065588704896, −8.542012710236359, −7.857163826001201, −7.384194432845308, −6.945796422403972, −6.221683133667762, −6.065507589573483, −5.358805929597528, −4.777550669362870, −4.387681569170842, −3.684125104030567, −3.270251738312324, −2.323858532487281, −1.872842405416270, −1.212292506653894, −0.2129838022907934, 0.2129838022907934, 1.212292506653894, 1.872842405416270, 2.323858532487281, 3.270251738312324, 3.684125104030567, 4.387681569170842, 4.777550669362870, 5.358805929597528, 6.065507589573483, 6.221683133667762, 6.945796422403972, 7.384194432845308, 7.857163826001201, 8.542012710236359, 8.947065588704896, 9.463527205873762, 10.00406172572555, 10.48031087921827, 10.95094167111918, 11.27834939642221, 11.92283081691368, 12.34449547499763, 12.65071124775741, 13.23813123508366

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