Properties

Label 2-9702-1.1-c1-0-60
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 − 1.41·5-s + 8-s − 1.41·10-s + 11-s + 2.82·13-s + 16-s + 5.65·17-s + 2.82·19-s − 1.41·20-s + 22-s + 2·23-s − 2.99·25-s + 2.82·26-s + 6·29-s − 1.41·31-s + 32-s + 5.65·34-s − 2·37-s + 2.82·38-s − 1.41·40-s − 5.65·41-s + 4·43-s + 44-s + 2·46-s − 4.24·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s + 0.301·11-s + 0.784·13-s + 0.250·16-s + 1.37·17-s + 0.648·19-s − 0.316·20-s + 0.213·22-s + 0.417·23-s − 0.599·25-s + 0.554·26-s + 1.11·29-s − 0.254·31-s + 0.176·32-s + 0.970·34-s − 0.328·37-s + 0.458·38-s − 0.223·40-s − 0.883·41-s + 0.609·43-s + 0.150·44-s + 0.294·46-s − 0.618·47-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

\(L(1)\) \(\approx\) \(3.481547169\)
\(L(\frac12)\) \(\approx\) \(3.481547169\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + 4.24T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−7.52439934963764878974639506515, −7.05320983189886319611053331847, −6.16945730938830495618789705776, −5.63771500837814560750558809974, −4.89129234534473132451797989846, −4.12104537381210029049728139367, −3.44978499480081786200908564789, −2.96492979434940412659366783048, −1.71120259922571551647729693970, −0.838545002784852569998979181643, 0.838545002784852569998979181643, 1.71120259922571551647729693970, 2.96492979434940412659366783048, 3.44978499480081786200908564789, 4.12104537381210029049728139367, 4.89129234534473132451797989846, 5.63771500837814560750558809974, 6.16945730938830495618789705776, 7.05320983189886319611053331847, 7.52439934963764878974639506515

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