Properties

Label 2-9702-1.1-c1-0-111
Degree $2$
Conductor $9702$
Sign $-1$
Analytic cond. $77.4708$
Root an. cond. $8.80175$
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  − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 2·13-s + 16-s + 5·17-s + 6·19-s − 20-s − 22-s − 7·23-s − 4·25-s + 2·26-s + 8·29-s − 10·31-s − 32-s − 5·34-s − 8·37-s − 6·38-s + 40-s + 7·41-s + 4·43-s + 44-s + 7·46-s − 47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.223·20-s − 0.213·22-s − 1.45·23-s − 4/5·25-s + 0.392·26-s + 1.48·29-s − 1.79·31-s − 0.176·32-s − 0.857·34-s − 1.31·37-s − 0.973·38-s + 0.158·40-s + 1.09·41-s + 0.609·43-s + 0.150·44-s + 1.03·46-s − 0.145·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9702\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(77.4708\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−7.36141660047017451180745820720, −7.02252447702095482585579797202, −5.84937030935396516375954294835, −5.56535267191391397009453512727, −4.49346556029279892548864256423, −3.66640971917477884603439082587, −3.03769457807762409156604606119, −2.00370142980636213080846097588, −1.12314680974055452083425111450, 0, 1.12314680974055452083425111450, 2.00370142980636213080846097588, 3.03769457807762409156604606119, 3.66640971917477884603439082587, 4.49346556029279892548864256423, 5.56535267191391397009453512727, 5.84937030935396516375954294835, 7.02252447702095482585579797202, 7.36141660047017451180745820720

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