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Properties

Label 2-96600-1.1-c1-0-51
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 − 4·11-s + 2·13-s + 2·17-s − 4·19-s − 21-s + 23-s + 27-s + 6·29-s + 8·31-s − 4·33-s + 2·37-s + 2·39-s − 6·41-s − 12·47-s + 49-s + 2·51-s + 2·53-s − 4·57-s − 8·59-s + 2·61-s − 63-s + 8·67-s + 69-s + 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s − 1.04·59-s + 0.256·61-s − 0.125·63-s + 0.977·67-s + 0.120·69-s + 0.949·71-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.85109536400514, −13.61101794492664, −13.02483970351342, −12.78954674763862, −12.10181157254591, −11.69159286656110, −10.90515531889622, −10.58050628595001, −9.949333025974449, −9.798991553530170, −8.949500415873758, −8.460066949709985, −8.082916494394389, −7.725577694012079, −6.869168213835132, −6.482534258211630, −6.013411745623598, −5.145030933348126, −4.834090198922859, −4.112724371616777, −3.474426225980061, −2.880200970245584, −2.507421771843930, −1.680614680519776, −0.9106514670596956, 0, 0.9106514670596956, 1.680614680519776, 2.507421771843930, 2.880200970245584, 3.474426225980061, 4.112724371616777, 4.834090198922859, 5.145030933348126, 6.013411745623598, 6.482534258211630, 6.869168213835132, 7.725577694012079, 8.082916494394389, 8.460066949709985, 8.949500415873758, 9.798991553530170, 9.949333025974449, 10.58050628595001, 10.90515531889622, 11.69159286656110, 12.10181157254591, 12.78954674763862, 13.02483970351342, 13.61101794492664, 13.85109536400514

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