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Properties

Label 2-96600-1.1-c1-0-15
Degree $2$
Conductor $96600$
Sign $1$
Analytic cond. $771.354$
Root an. cond. $27.7732$
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 − 7-s + 9-s − 2·13-s − 6·17-s + 4·19-s + 21-s + 23-s − 27-s + 6·29-s + 4·31-s − 2·37-s + 2·39-s + 10·41-s + 8·43-s − 4·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s − 4·59-s + 2·61-s − 63-s + 8·67-s − 69-s − 8·71-s − 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s + 0.977·67-s − 0.120·69-s − 0.949·71-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(771.354\)
Root analytic conductor: \(27.7732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.540545423\)
\(L(\frac12)\) \(\approx\) \(1.540545423\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.76547706019151, −13.23414834618474, −12.80117490985863, −12.26802432933265, −11.89501283631147, −11.25289368570860, −10.98777266188854, −10.29861354489186, −9.893977137107513, −9.407831117121713, −8.811327945187431, −8.411918520661208, −7.507901627986556, −7.295419776687123, −6.605006943640446, −6.233345456996254, −5.598351272436571, −5.077554655976761, −4.300179324564441, −4.242094017307589, −3.106618141907260, −2.707684810364705, −2.007532564885235, −1.087244441368220, −0.4596688515119279, 0.4596688515119279, 1.087244441368220, 2.007532564885235, 2.707684810364705, 3.106618141907260, 4.242094017307589, 4.300179324564441, 5.077554655976761, 5.598351272436571, 6.233345456996254, 6.605006943640446, 7.295419776687123, 7.507901627986556, 8.411918520661208, 8.811327945187431, 9.407831117121713, 9.893977137107513, 10.29861354489186, 10.98777266188854, 11.25289368570860, 11.89501283631147, 12.26802432933265, 12.80117490985863, 13.23414834618474, 13.76547706019151

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