Properties

Label 2-9702-1.1-c1-0-48
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 − 2.80·5-s − 8-s + 2.80·10-s + 11-s + 16-s + 7.96·17-s + 6.48·19-s − 2.80·20-s − 22-s + 5.28·23-s + 2.87·25-s − 5.12·29-s + 8.96·31-s − 32-s − 7.96·34-s + 10.4·37-s − 6.48·38-s + 2.80·40-s − 5.09·41-s + 11.4·43-s + 44-s − 5.28·46-s + 0.322·47-s − 2.87·50-s − 8·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.25·5-s − 0.353·8-s + 0.887·10-s + 0.301·11-s + 0.250·16-s + 1.93·17-s + 1.48·19-s − 0.627·20-s − 0.213·22-s + 1.10·23-s + 0.574·25-s − 0.952·29-s + 1.61·31-s − 0.176·32-s − 1.36·34-s + 1.72·37-s − 1.05·38-s + 0.443·40-s − 0.795·41-s + 1.74·43-s + 0.150·44-s − 0.779·46-s + 0.0471·47-s − 0.406·50-s − 1.09·53-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: 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

\(L(1)\) \(\approx\) \(1.401588979\)
\(L(\frac12)\) \(\approx\) \(1.401588979\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.80T + 5T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
19 \( 1 - 6.48T + 19T^{2} \)
23 \( 1 - 5.28T + 23T^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 0.322T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + 0.871T + 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 + 7.09T + 67T^{2} \)
71 \( 1 - 7.44T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 6.80T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 - 1.61T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−7.74058182651774404112913926360, −7.38160164959207173344265217663, −6.46838428320315215235015269043, −5.69192670022197182700722860288, −4.94525939477218560836992475006, −4.04890950919402590687095102181, −3.29043204129025094557510511936, −2.79646670768548651461284563831, −1.28994298532976653712006844945, −0.73283151413975283700636296459, 0.73283151413975283700636296459, 1.28994298532976653712006844945, 2.79646670768548651461284563831, 3.29043204129025094557510511936, 4.04890950919402590687095102181, 4.94525939477218560836992475006, 5.69192670022197182700722860288, 6.46838428320315215235015269043, 7.38160164959207173344265217663, 7.74058182651774404112913926360

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