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Properties

Label 2-10800-1.1-c1-0-21
Degree $2$
Conductor $10800$
Sign $1$
Analytic cond. $86.2384$
Root an. cond. $9.28646$
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  − 7-s − 2·13-s + 7·19-s + 7·31-s + 37-s + 5·43-s − 6·49-s − 61-s − 16·67-s − 17·73-s + 13·79-s + 2·91-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.554·13-s + 1.60·19-s + 1.25·31-s + 0.164·37-s + 0.762·43-s − 6/7·49-s − 0.128·61-s − 1.95·67-s − 1.98·73-s + 1.46·79-s + 0.209·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(86.2384\)
Root analytic conductor: \(9.28646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.857248631\)
\(L(\frac12)\) \(\approx\) \(1.857248631\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−16.42756229176722, −15.96431925862619, −15.52548256858014, −14.71365125905493, −14.34079497655688, −13.49719247335045, −13.31068932655625, −12.35400516529026, −11.96322391021624, −11.45187161284184, −10.60475615388235, −10.08283196318456, −9.464875390895734, −9.032266017180557, −8.099225919241881, −7.592654407494679, −6.973351600212710, −6.236485871794988, −5.589873862992146, −4.857945517958290, −4.206212174805879, −3.193584808430966, −2.772468513379515, −1.641052835630650, −0.6455309053882077, 0.6455309053882077, 1.641052835630650, 2.772468513379515, 3.193584808430966, 4.206212174805879, 4.857945517958290, 5.589873862992146, 6.236485871794988, 6.973351600212710, 7.592654407494679, 8.099225919241881, 9.032266017180557, 9.464875390895734, 10.08283196318456, 10.60475615388235, 11.45187161284184, 11.96322391021624, 12.35400516529026, 13.31068932655625, 13.49719247335045, 14.34079497655688, 14.71365125905493, 15.52548256858014, 15.96431925862619, 16.42756229176722

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