Properties

Label 2-161700-1.1-c1-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 11-s + 2·13-s − 6·17-s − 8·19-s + 6·23-s + 27-s + 6·29-s − 2·31-s − 33-s − 2·37-s + 2·39-s − 8·43-s − 12·47-s − 6·51-s − 6·53-s − 8·57-s − 6·59-s − 8·61-s − 2·67-s + 6·69-s − 10·73-s + 8·79-s + 81-s + 12·83-s + 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.25·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s − 1.21·43-s − 1.75·47-s − 0.840·51-s − 0.824·53-s − 1.05·57-s − 0.781·59-s − 1.02·61-s − 0.244·67-s + 0.722·69-s − 1.17·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + 0.643·87-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

\(L(1)\) \(\approx\) \(1.346873066\)
\(L(\frac12)\) \(\approx\) \(1.346873066\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.40583039372316, −12.83993898745791, −12.53824941084202, −11.84789710155342, −11.20542984139730, −10.87989312419870, −10.50788266757971, −9.944023432839283, −9.304034739647118, −8.796534312348496, −8.583907159155280, −8.082567495882920, −7.510197749537886, −6.749269054325920, −6.530172745621215, −6.128422397755929, −5.167865034895745, −4.677727463862687, −4.390190608553040, −3.606901031735367, −3.087634729719712, −2.537059458117174, −1.867544726653304, −1.408069144063876, −0.3085046352696283, 0.3085046352696283, 1.408069144063876, 1.867544726653304, 2.537059458117174, 3.087634729719712, 3.606901031735367, 4.390190608553040, 4.677727463862687, 5.167865034895745, 6.128422397755929, 6.530172745621215, 6.749269054325920, 7.510197749537886, 8.082567495882920, 8.583907159155280, 8.796534312348496, 9.304034739647118, 9.944023432839283, 10.50788266757971, 10.87989312419870, 11.20542984139730, 11.84789710155342, 12.53824941084202, 12.83993898745791, 13.40583039372316

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