Properties

Label 2-9702-1.1-c1-0-43
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 + 6·23-s − 25-s + 2·26-s + 6·29-s − 2·31-s + 32-s − 6·34-s + 2·37-s + 4·38-s − 2·40-s − 10·41-s − 6·43-s + 44-s + 6·46-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 + 1.25·23-s − 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.316·40-s − 1.56·41-s − 0.914·43-s + 0.150·44-s + 0.884·46-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.770938535\)
\(L(\frac12)\) \(\approx\) \(2.770938535\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−7.54824232068911395656064191370, −6.74805492397303772246489990919, −6.58364041888834181199164421811, −5.42804619739000563492874352692, −4.89526761664556889159659664564, −4.13729580279245842571558267901, −3.54695301666708082056930384176, −2.85146475476105833219768694809, −1.82174119142586825272219543863, −0.71419354043948938817251142086, 0.71419354043948938817251142086, 1.82174119142586825272219543863, 2.85146475476105833219768694809, 3.54695301666708082056930384176, 4.13729580279245842571558267901, 4.89526761664556889159659664564, 5.42804619739000563492874352692, 6.58364041888834181199164421811, 6.74805492397303772246489990919, 7.54824232068911395656064191370

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