Properties

Label 2-9702-1.1-c1-0-54
Degree $2$
Conductor $9702$
Sign $1$
Analytic cond. $77.4708$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4.24·5-s − 8-s + 4.24·10-s − 11-s + 5.65·13-s + 16-s + 7.07·17-s + 1.41·19-s − 4.24·20-s + 22-s + 8·23-s + 12.9·25-s − 5.65·26-s + 8·29-s + 4.24·31-s − 32-s − 7.07·34-s + 2·37-s − 1.41·38-s + 4.24·40-s − 1.41·41-s + 8·43-s − 44-s − 8·46-s − 9.89·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.89·5-s − 0.353·8-s + 1.34·10-s − 0.301·11-s + 1.56·13-s + 0.250·16-s + 1.71·17-s + 0.324·19-s − 0.948·20-s + 0.213·22-s + 1.66·23-s + 2.59·25-s − 1.10·26-s + 1.48·29-s + 0.762·31-s − 0.176·32-s − 1.21·34-s + 0.328·37-s − 0.229·38-s + 0.670·40-s − 0.220·41-s + 1.21·43-s − 0.150·44-s − 1.17·46-s − 1.44·47-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: no
Arithmetic: yes
Character: $\chi_{9702} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.342924979\)
\(L(\frac12)\) \(\approx\) \(1.342924979\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.24T + 5T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 - 1.41T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.921257870560189945119898077078, −7.13838279808742192183071174910, −6.62897903441791918434694929995, −5.68337931795868516778550008780, −4.85206790392352138464152111370, −4.05457990486977706668906508931, −3.23637832527189334142074238518, −2.95218741131861289437367578106, −1.15670336413532022742848020324, −0.77744458792107800387524078733, 0.77744458792107800387524078733, 1.15670336413532022742848020324, 2.95218741131861289437367578106, 3.23637832527189334142074238518, 4.05457990486977706668906508931, 4.85206790392352138464152111370, 5.68337931795868516778550008780, 6.62897903441791918434694929995, 7.13838279808742192183071174910, 7.921257870560189945119898077078

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