Properties

Label 2-9702-1.1-c1-0-30
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 11-s − 2·13-s + 16-s − 6·17-s + 4·19-s − 2·20-s − 22-s + 4·23-s − 25-s − 2·26-s − 2·29-s + 4·31-s + 32-s − 6·34-s − 2·37-s + 4·38-s − 2·40-s − 6·41-s − 44-s + 4·46-s − 8·47-s − 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.316·40-s − 0.937·41-s − 0.150·44-s + 0.589·46-s − 1.16·47-s − 0.141·50-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: $\chi_{9702} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.148400901\)
\(L(\frac12)\) \(\approx\) \(2.148400901\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.31524169886963257395601471545, −7.17392354889171205971358987767, −6.36063463733747478696312824906, −5.41505120623238970032232815306, −4.90279695193449377134128499565, −4.19984568778577016705058800267, −3.53143401028865361989935683360, −2.75348738236178944457442835036, −1.95056468130131064241502431444, −0.60404099945041688911466731367, 0.60404099945041688911466731367, 1.95056468130131064241502431444, 2.75348738236178944457442835036, 3.53143401028865361989935683360, 4.19984568778577016705058800267, 4.90279695193449377134128499565, 5.41505120623238970032232815306, 6.36063463733747478696312824906, 7.17392354889171205971358987767, 7.31524169886963257395601471545

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