Properties

Label 2-100800-1.1-c1-0-233
Degree $2$
Conductor $100800$
Sign $-1$
Analytic cond. $804.892$
Root an. cond. $28.3706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 6·11-s + 2·13-s + 2·17-s + 4·19-s − 4·23-s + 2·29-s + 2·31-s − 10·37-s − 6·41-s + 2·43-s + 2·47-s + 49-s + 6·53-s + 4·59-s + 12·61-s − 10·67-s + 12·71-s + 2·73-s + 6·77-s + 16·79-s + 12·83-s − 14·89-s − 2·91-s − 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 0.371·29-s + 0.359·31-s − 1.64·37-s − 0.937·41-s + 0.304·43-s + 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.53·61-s − 1.22·67-s + 1.42·71-s + 0.234·73-s + 0.683·77-s + 1.80·79-s + 1.31·83-s − 1.48·89-s − 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(804.892\)
Root analytic conductor: \(28.3706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.78774726548900, −13.61317594944813, −13.08426378693197, −12.49804602560414, −12.06607762599689, −11.68638784287625, −10.82572247894190, −10.60814782088973, −10.04123989615840, −9.726575477287270, −9.022789670901755, −8.363387157700990, −8.057793338595005, −7.572065750911044, −6.896292109638312, −6.499830601568692, −5.652080788646434, −5.323520033228944, −4.993829526770624, −3.960558509418045, −3.617163088656679, −2.856998790102373, −2.453824875447625, −1.634844397108997, −0.7853413712040585, 0, 0.7853413712040585, 1.634844397108997, 2.453824875447625, 2.856998790102373, 3.617163088656679, 3.960558509418045, 4.993829526770624, 5.323520033228944, 5.652080788646434, 6.499830601568692, 6.896292109638312, 7.572065750911044, 8.057793338595005, 8.363387157700990, 9.022789670901755, 9.726575477287270, 10.04123989615840, 10.60814782088973, 10.82572247894190, 11.68638784287625, 12.06607762599689, 12.49804602560414, 13.08426378693197, 13.61317594944813, 13.78774726548900

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