Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 5·11-s − 2·13-s − 4·17-s + 19-s − 21-s − 23-s + 27-s − 4·29-s + 2·31-s − 5·33-s + 10·37-s − 2·39-s + 5·41-s − 8·43-s + 7·47-s + 49-s − 4·51-s − 9·53-s + 57-s + 4·59-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 − 1.50·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.870·33-s + 1.64·37-s − 0.320·39-s + 0.780·41-s − 1.21·43-s + 1.02·47-s + 1/7·49-s − 0.560·51-s − 1.23·53-s + 0.132·57-s + 0.520·59-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: \(0\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9389330484\)
\(L(\frac12)\) \(\approx\) \(0.9389330484\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.50152102038595, −13.37709121714546, −12.97520151030274, −12.47347922889907, −11.86957287871134, −11.24912986076041, −10.84459072838619, −10.22084440318814, −9.827889979358413, −9.392721193941437, −8.781726790425449, −8.255886126266111, −7.794700012419564, −7.288998291481473, −6.880023753878999, −6.061163373302228, −5.657281525277958, −4.975541929780832, −4.402052391774419, −3.959498288650867, −2.976454835902775, −2.738910718173420, −2.180929126410355, −1.355921106322460, −0.2843234748651964, 0.2843234748651964, 1.355921106322460, 2.180929126410355, 2.738910718173420, 2.976454835902775, 3.959498288650867, 4.402052391774419, 4.975541929780832, 5.657281525277958, 6.061163373302228, 6.880023753878999, 7.288998291481473, 7.794700012419564, 8.255886126266111, 8.781726790425449, 9.392721193941437, 9.827889979358413, 10.22084440318814, 10.84459072838619, 11.24912986076041, 11.86957287871134, 12.47347922889907, 12.97520151030274, 13.37709121714546, 13.50152102038595

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