Properties

Label 2-96600-1.1-c1-0-49
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 + 2·11-s − 6·13-s + 2·19-s − 21-s − 23-s + 27-s − 6·29-s − 2·31-s + 2·33-s + 12·37-s − 6·39-s + 2·41-s + 8·43-s − 12·47-s + 49-s − 6·53-s + 2·57-s − 10·61-s − 63-s − 4·67-s − 69-s − 6·71-s − 2·73-s − 2·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s + 1.97·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.264·57-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s − 0.712·71-s − 0.234·73-s − 0.227·77-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.14251853882071, −13.58075808110809, −12.92437591993370, −12.71457386466391, −12.16862136340035, −11.54035380889254, −11.24040318244608, −10.46270204354956, −9.919727033161313, −9.475191637513621, −9.271079680362013, −8.662542036123472, −7.775974884039485, −7.587033668271800, −7.194947261524310, −6.324901243083288, −6.052125916477992, −5.249801782571133, −4.605580641198957, −4.266117791088401, −3.407928273818656, −3.034884657140726, −2.294086908456759, −1.806495903206662, −0.8790132887945035, 0, 0.8790132887945035, 1.806495903206662, 2.294086908456759, 3.034884657140726, 3.407928273818656, 4.266117791088401, 4.605580641198957, 5.249801782571133, 6.052125916477992, 6.324901243083288, 7.194947261524310, 7.587033668271800, 7.775974884039485, 8.662542036123472, 9.271079680362013, 9.475191637513621, 9.919727033161313, 10.46270204354956, 11.24040318244608, 11.54035380889254, 12.16862136340035, 12.71457386466391, 12.92437591993370, 13.58075808110809, 14.14251853882071

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