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Properties

Label 2-96600-1.1-c1-0-44
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 − 6·11-s + 4·13-s − 2·17-s + 4·19-s − 21-s − 23-s − 27-s + 2·29-s − 2·31-s + 6·33-s − 12·37-s − 4·39-s + 6·41-s + 4·43-s + 4·47-s + 49-s + 2·51-s − 12·53-s − 4·57-s + 10·59-s − 10·61-s + 63-s + 8·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 1.04·33-s − 1.97·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s − 1.64·53-s − 0.529·57-s + 1.30·59-s − 1.28·61-s + 0.125·63-s + 0.977·67-s + 0.120·69-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.13394329395445, −13.41196620338226, −13.14127421184066, −12.48246389489326, −12.20682166343532, −11.39222768228354, −11.09230689221015, −10.66004599892137, −10.26393513044424, −9.687561342346890, −9.001348742646886, −8.516591508986625, −7.995586450848132, −7.484439870361617, −7.058390012260290, −6.320005477193751, −5.756242090272907, −5.383692130892786, −4.858057099614756, −4.304884784879035, −3.536370783532133, −2.996984729093612, −2.252584027447567, −1.598753750715911, −0.7997171177347311, 0, 0.7997171177347311, 1.598753750715911, 2.252584027447567, 2.996984729093612, 3.536370783532133, 4.304884784879035, 4.858057099614756, 5.383692130892786, 5.756242090272907, 6.320005477193751, 7.058390012260290, 7.484439870361617, 7.995586450848132, 8.516591508986625, 9.001348742646886, 9.687561342346890, 10.26393513044424, 10.66004599892137, 11.09230689221015, 11.39222768228354, 12.20682166343532, 12.48246389489326, 13.14127421184066, 13.41196620338226, 14.13394329395445

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