Properties

Label 2-212160-1.1-c1-0-20
Degree $2$
Conductor $212160$
Sign $1$
Analytic cond. $1694.10$
Root an. cond. $41.1595$
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 − 5-s + 9-s + 4·11-s − 13-s − 15-s + 17-s − 4·19-s + 25-s + 27-s + 2·29-s + 4·33-s − 6·37-s − 39-s − 6·41-s − 4·43-s − 45-s − 7·49-s + 51-s − 6·53-s − 4·55-s − 4·57-s + 4·59-s − 6·61-s + 65-s + 12·67-s + 16·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.140·51-s − 0.824·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.124·65-s + 1.46·67-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(212160\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1694.10\)
Root analytic conductor: \(41.1595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 212160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.380175475\)
\(L(\frac12)\) \(\approx\) \(2.380175475\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.91206814488558, −12.47953443008461, −12.20113759890354, −11.57330658208670, −11.25923881718552, −10.62549759369110, −10.17441000449070, −9.610110282718952, −9.287576780016405, −8.655668247958884, −8.282928323643910, −7.984684373258024, −7.143134578707913, −6.881525687470080, −6.407122812462980, −5.866235916250572, −4.975993526485269, −4.745231287877953, −4.038380479268979, −3.539202405626876, −3.241740871166513, −2.384183584936923, −1.832616949236815, −1.271105734402625, −0.4185159062344675, 0.4185159062344675, 1.271105734402625, 1.832616949236815, 2.384183584936923, 3.241740871166513, 3.539202405626876, 4.038380479268979, 4.745231287877953, 4.975993526485269, 5.866235916250572, 6.407122812462980, 6.881525687470080, 7.143134578707913, 7.984684373258024, 8.282928323643910, 8.655668247958884, 9.287576780016405, 9.610110282718952, 10.17441000449070, 10.62549759369110, 11.25923881718552, 11.57330658208670, 12.20113759890354, 12.47953443008461, 12.91206814488558

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