Properties

Label 2-9702-1.1-c1-0-53
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 + 1.79·5-s − 8-s − 1.79·10-s − 11-s + 3.63·13-s + 16-s − 6.11·17-s + 1.84·19-s + 1.79·20-s + 22-s + 7.37·23-s − 1.76·25-s − 3.63·26-s + 10.4·29-s + 7.97·31-s − 32-s + 6.11·34-s + 10.1·37-s − 1.84·38-s − 1.79·40-s − 8.17·41-s − 2.28·43-s − 44-s − 7.37·46-s + 0.669·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.804·5-s − 0.353·8-s − 0.568·10-s − 0.301·11-s + 1.00·13-s + 0.250·16-s − 1.48·17-s + 0.422·19-s + 0.402·20-s + 0.213·22-s + 1.53·23-s − 0.353·25-s − 0.713·26-s + 1.93·29-s + 1.43·31-s − 0.176·32-s + 1.04·34-s + 1.67·37-s − 0.298·38-s − 0.284·40-s − 1.27·41-s − 0.348·43-s − 0.150·44-s − 1.08·46-s + 0.0976·47-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

\(L(1)\) \(\approx\) \(1.903187674\)
\(L(\frac12)\) \(\approx\) \(1.903187674\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.79T + 5T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
17 \( 1 + 6.11T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 + 8.17T + 41T^{2} \)
43 \( 1 + 2.28T + 43T^{2} \)
47 \( 1 - 0.669T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 0.274T + 61T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 5.27T + 83T^{2} \)
89 \( 1 + 1.98T + 89T^{2} \)
97 \( 1 - 4.12T + 97T^{2} \)
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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

−7.83530269542118075991057960451, −6.77025385224389762051470019992, −6.55179677144810455241336471930, −5.83023057328940260415827163622, −4.95166239804956323136239028614, −4.30128557941788803465807220847, −3.07274148683137598650688776636, −2.57349429624634832332477361136, −1.57100030283287942985611410501, −0.76948084543206307722149353370, 0.76948084543206307722149353370, 1.57100030283287942985611410501, 2.57349429624634832332477361136, 3.07274148683137598650688776636, 4.30128557941788803465807220847, 4.95166239804956323136239028614, 5.83023057328940260415827163622, 6.55179677144810455241336471930, 6.77025385224389762051470019992, 7.83530269542118075991057960451

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