Properties

Label 2-96600-1.1-c1-0-23
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 − 4·11-s + 2·17-s + 2·19-s + 21-s − 23-s − 27-s + 8·31-s + 4·33-s + 6·37-s − 8·41-s + 8·43-s + 49-s − 2·51-s + 6·53-s − 2·57-s − 2·59-s − 2·61-s − 63-s + 8·67-s + 69-s + 10·71-s − 6·73-s + 4·77-s + 14·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.485·17-s + 0.458·19-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 − 1.24·41-s + 1.21·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.264·57-s − 0.260·59-s − 0.256·61-s − 0.125·63-s + 0.977·67-s + 0.120·69-s + 1.18·71-s − 0.702·73-s + 0.455·77-s + 1.57·79-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

\(L(1)\) \(\approx\) \(1.705104141\)
\(L(\frac12)\) \(\approx\) \(1.705104141\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.68679846940037, −13.34056089247253, −12.70086284371104, −12.37927181728946, −11.85934387130292, −11.33681778312718, −10.83647574372731, −10.31373567866242, −9.872308528110259, −9.586886825636826, −8.715050249581223, −8.289815321666461, −7.586260505631263, −7.409612317584143, −6.543060057049813, −6.172889649471300, −5.587123440128912, −5.070580454842922, −4.625445343176332, −3.879189411674468, −3.249126764132119, −2.629773873657866, −2.061789429078202, −1.026434988111728, −0.5041701605467070, 0.5041701605467070, 1.026434988111728, 2.061789429078202, 2.629773873657866, 3.249126764132119, 3.879189411674468, 4.625445343176332, 5.070580454842922, 5.587123440128912, 6.172889649471300, 6.543060057049813, 7.409612317584143, 7.586260505631263, 8.289815321666461, 8.715050249581223, 9.586886825636826, 9.872308528110259, 10.31373567866242, 10.83647574372731, 11.33681778312718, 11.85934387130292, 12.37927181728946, 12.70086284371104, 13.34056089247253, 13.68679846940037

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