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Properties

Label 2-96600-1.1-c1-0-52
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s + 3·11-s + 2·13-s − 4·17-s + 3·19-s + 21-s + 23-s − 27-s − 2·31-s − 3·33-s + 10·37-s − 2·39-s + 7·41-s + 4·43-s − 3·47-s + 49-s + 4·51-s − 53-s − 3·57-s + 4·59-s − 6·61-s − 63-s − 69-s − 12·71-s + 16·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.554·13-s − 0.970·17-s + 0.688·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.359·31-s − 0.522·33-s + 1.64·37-s − 0.320·39-s + 1.09·41-s + 0.609·43-s − 0.437·47-s + 1/7·49-s + 0.560·51-s − 0.137·53-s − 0.397·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.120·69-s − 1.42·71-s + 1.87·73-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: \(1\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.08845166002401, −13.40017573814376, −13.04219707743063, −12.63651386569195, −11.96854850912099, −11.59190632226942, −11.01159639851698, −10.86902701702466, −10.03509071364408, −9.504063310149038, −9.214772903478228, −8.651480187248130, −7.974475237050976, −7.428761006469787, −6.799620414284525, −6.493000137915837, −5.814887120807179, −5.558843001127304, −4.571654496490738, −4.287612602804204, −3.685215587921686, −2.958172678575351, −2.332639717210001, −1.418218479114334, −0.9308737421981259, 0, 0.9308737421981259, 1.418218479114334, 2.332639717210001, 2.958172678575351, 3.685215587921686, 4.287612602804204, 4.571654496490738, 5.558843001127304, 5.814887120807179, 6.493000137915837, 6.799620414284525, 7.428761006469787, 7.974475237050976, 8.651480187248130, 9.214772903478228, 9.504063310149038, 10.03509071364408, 10.86902701702466, 11.01159639851698, 11.59190632226942, 11.96854850912099, 12.63651386569195, 13.04219707743063, 13.40017573814376, 14.08845166002401

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