Properties

Label 2-9702-1.1-c1-0-26
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 − 3·5-s − 8-s + 3·10-s + 11-s + 2·13-s + 16-s + 3·17-s + 2·19-s − 3·20-s − 22-s − 3·23-s + 4·25-s − 2·26-s + 2·31-s − 32-s − 3·34-s + 8·37-s − 2·38-s + 3·40-s + 9·41-s − 4·43-s + 44-s + 3·46-s − 3·47-s − 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 0.670·20-s − 0.213·22-s − 0.625·23-s + 4/5·25-s − 0.392·26-s + 0.359·31-s − 0.176·32-s − 0.514·34-s + 1.31·37-s − 0.324·38-s + 0.474·40-s + 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.442·46-s − 0.437·47-s − 0.565·50-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

\(L(1)\) \(\approx\) \(1.035068702\)
\(L(\frac12)\) \(\approx\) \(1.035068702\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−7.70568396368074032136470678510, −7.32475867031080814209612554549, −6.39371790783118345004403033853, −5.85107407193854462317294306387, −4.81908334791639931418960930332, −4.03423157696245935221029134224, −3.45465563523040557671191999558, −2.64600941070577403570016606989, −1.42905787877392315938432351420, −0.58025390721148013899741029319, 0.58025390721148013899741029319, 1.42905787877392315938432351420, 2.64600941070577403570016606989, 3.45465563523040557671191999558, 4.03423157696245935221029134224, 4.81908334791639931418960930332, 5.85107407193854462317294306387, 6.39371790783118345004403033853, 7.32475867031080814209612554549, 7.70568396368074032136470678510

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