Properties

Label 2-9702-1.1-c1-0-74
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 + 4·13-s + 16-s − 4·19-s + 2·20-s − 22-s − 4·23-s − 25-s + 4·26-s − 2·29-s + 10·31-s + 32-s − 6·37-s − 4·38-s + 2·40-s − 4·43-s − 44-s − 4·46-s + 10·47-s − 50-s + 4·52-s + 14·53-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 + 1.10·13-s + 1/4·16-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 0.784·26-s − 0.371·29-s + 1.79·31-s + 0.176·32-s − 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.609·43-s − 0.150·44-s − 0.589·46-s + 1.45·47-s − 0.141·50-s + 0.554·52-s + 1.92·53-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

\(L(1)\) \(\approx\) \(4.259558353\)
\(L(\frac12)\) \(\approx\) \(4.259558353\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.60662678357074446672630692097, −6.66165493167683007748095504732, −6.28710805779090032434203569574, −5.62199336160611525863427043947, −5.06823717564074315586051720075, −4.06867461412619442474056297162, −3.63174413599045265668328472044, −2.45096584239443641859321352226, −2.03357682397592720278356318451, −0.890566128143998534104277981591, 0.890566128143998534104277981591, 2.03357682397592720278356318451, 2.45096584239443641859321352226, 3.63174413599045265668328472044, 4.06867461412619442474056297162, 5.06823717564074315586051720075, 5.62199336160611525863427043947, 6.28710805779090032434203569574, 6.66165493167683007748095504732, 7.60662678357074446672630692097

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