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Properties

Label 2-199920-1.1-c1-0-78
Degree $2$
Conductor $199920$
Sign $1$
Analytic cond. $1596.36$
Root an. cond. $39.9545$
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 + 6·11-s + 4·13-s + 15-s + 17-s − 2·19-s + 2·23-s + 25-s − 27-s − 4·29-s − 6·33-s − 2·37-s − 4·39-s − 6·41-s − 4·43-s − 45-s + 4·47-s − 51-s + 6·53-s − 6·55-s + 2·57-s + 8·59-s + 10·61-s − 4·65-s − 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 0.258·15-s + 0.242·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.04·33-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 0.140·51-s + 0.824·53-s − 0.809·55-s + 0.264·57-s + 1.04·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(199920\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1596.36\)
Root analytic conductor: \(39.9545\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 199920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.767326330\)
\(L(\frac12)\) \(\approx\) \(2.767326330\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
17 \( 1 - T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.05819743963320, −12.40433743407937, −12.06581963814223, −11.68946912611075, −11.23681305511859, −10.80706082860365, −10.43095401869377, −9.654593410498875, −9.314692358220571, −8.821403601654483, −8.319367937394864, −7.911162478277878, −7.090858829593998, −6.645230906372022, −6.546107714681712, −5.771255236463298, −5.321342299367551, −4.727886396260463, −4.005781581144150, −3.704513631143893, −3.401718552392788, −2.294148831155949, −1.717118599821240, −1.029083321799857, −0.5876761361641106, 0.5876761361641106, 1.029083321799857, 1.717118599821240, 2.294148831155949, 3.401718552392788, 3.704513631143893, 4.005781581144150, 4.727886396260463, 5.321342299367551, 5.771255236463298, 6.546107714681712, 6.645230906372022, 7.090858829593998, 7.911162478277878, 8.319367937394864, 8.821403601654483, 9.314692358220571, 9.654593410498875, 10.43095401869377, 10.80706082860365, 11.23681305511859, 11.68946912611075, 12.06581963814223, 12.40433743407937, 13.05819743963320

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