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Properties

Label 2-96600-1.1-c1-0-45
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 + 4·13-s + 6·17-s − 6·19-s − 21-s − 23-s − 27-s − 8·31-s + 4·33-s − 6·37-s − 4·39-s + 12·41-s − 8·43-s + 49-s − 6·51-s + 6·53-s + 6·57-s + 6·59-s − 6·61-s + 63-s − 8·67-s + 69-s + 6·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 1.45·17-s − 1.37·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s + 1.87·41-s − 1.21·43-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.794·57-s + 0.781·59-s − 0.768·61-s + 0.125·63-s − 0.977·67-s + 0.120·69-s + 0.712·71-s − 0.234·73-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.98154348030851, −13.45234011455648, −13.01449543047672, −12.55673500357836, −12.15655233033566, −11.52924390047804, −10.96833972903349, −10.63074811969099, −10.32409190708389, −9.685728108101487, −8.990542167363122, −8.511586728972587, −7.989361243714823, −7.539029124618551, −7.030292330890148, −6.188966484080048, −5.945341498543277, −5.258344219446240, −4.997664863242735, −4.083524211966597, −3.714019685641643, −2.992917526510731, −2.185703679356392, −1.616891292478251, −0.8247331161067594, 0, 0.8247331161067594, 1.616891292478251, 2.185703679356392, 2.992917526510731, 3.714019685641643, 4.083524211966597, 4.997664863242735, 5.258344219446240, 5.945341498543277, 6.188966484080048, 7.030292330890148, 7.539029124618551, 7.989361243714823, 8.511586728972587, 8.990542167363122, 9.685728108101487, 10.32409190708389, 10.63074811969099, 10.96833972903349, 11.52924390047804, 12.15655233033566, 12.55673500357836, 13.01449543047672, 13.45234011455648, 13.98154348030851

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