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Properties

Label 2-96600-1.1-c1-0-4
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 + 6·17-s − 19-s + 21-s − 23-s − 27-s + 8·29-s − 6·31-s − 8·37-s − 6·41-s + 6·43-s + 3·47-s + 49-s − 6·51-s − 6·53-s + 57-s − 13·59-s − 13·61-s − 63-s − 10·67-s + 69-s − 8·71-s − 8·73-s + 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.45·17-s − 0.229·19-s + 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.48·29-s − 1.07·31-s − 1.31·37-s − 0.937·41-s + 0.914·43-s + 0.437·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.132·57-s − 1.69·59-s − 1.66·61-s − 0.125·63-s − 1.22·67-s + 0.120·69-s − 0.949·71-s − 0.936·73-s + 0.900·79-s + 1/9·81-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

\(L(1)\) \(\approx\) \(0.9896491584\)
\(L(\frac12)\) \(\approx\) \(0.9896491584\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.79598158375904, −13.36266558513259, −12.49931208765566, −12.34453216240525, −12.04691526848775, −11.34415293665528, −10.72087331216365, −10.38740633216086, −9.993400111421070, −9.325285252101101, −8.900376669869685, −8.271813076837907, −7.617522964880925, −7.300676840681181, −6.613980650066984, −6.101648238354185, −5.666024139705628, −5.102147898919536, −4.507918482771102, −3.934632560956737, −3.132092350615535, −2.893498265050276, −1.712300014575191, −1.339999259207989, −0.3336170675168563, 0.3336170675168563, 1.339999259207989, 1.712300014575191, 2.893498265050276, 3.132092350615535, 3.934632560956737, 4.507918482771102, 5.102147898919536, 5.666024139705628, 6.101648238354185, 6.613980650066984, 7.300676840681181, 7.617522964880925, 8.271813076837907, 8.900376669869685, 9.325285252101101, 9.993400111421070, 10.38740633216086, 10.72087331216365, 11.34415293665528, 12.04691526848775, 12.34453216240525, 12.49931208765566, 13.36266558513259, 13.79598158375904

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