Properties

Label 2-9702-1.1-c1-0-135
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2.74·5-s − 8-s − 2.74·10-s − 11-s − 5.49·13-s + 16-s + 17-s + 8.03·19-s + 2.74·20-s + 22-s + 1.29·23-s + 2.54·25-s + 5.49·26-s − 4.54·29-s − 5.08·31-s − 32-s − 34-s − 4.03·37-s − 8.03·38-s − 2.74·40-s + 5.54·41-s − 4.03·43-s − 44-s − 1.29·46-s − 9.29·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.22·5-s − 0.353·8-s − 0.868·10-s − 0.301·11-s − 1.52·13-s + 0.250·16-s + 0.242·17-s + 1.84·19-s + 0.614·20-s + 0.213·22-s + 0.269·23-s + 0.508·25-s + 1.07·26-s − 0.843·29-s − 0.913·31-s − 0.176·32-s − 0.171·34-s − 0.663·37-s − 1.30·38-s − 0.434·40-s + 0.865·41-s − 0.615·43-s − 0.150·44-s − 0.190·46-s − 1.35·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 5T^{2} \)
13 \( 1 + 5.49T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 8.03T + 19T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 + 4.54T + 29T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + 4.03T + 37T^{2} \)
41 \( 1 - 5.54T + 41T^{2} \)
43 \( 1 + 4.03T + 43T^{2} \)
47 \( 1 + 9.29T + 47T^{2} \)
53 \( 1 + 5.49T + 53T^{2} \)
59 \( 1 + 9.52T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 - 3.54T + 67T^{2} \)
71 \( 1 + 2.54T + 71T^{2} \)
73 \( 1 - 8.58T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + 6.50T + 89T^{2} \)
97 \( 1 - 0.0872T + 97T^{2} \)
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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

−7.52549781412875013337099053754, −6.77585713292461089085449624996, −5.99478621779306996641276815298, −5.25307533733522518018502556836, −4.97576629217043130456938403403, −3.54326963820004365151583007326, −2.80186771278151858170541182136, −2.04472049630453939729398549057, −1.31480939060367634484582915095, 0, 1.31480939060367634484582915095, 2.04472049630453939729398549057, 2.80186771278151858170541182136, 3.54326963820004365151583007326, 4.97576629217043130456938403403, 5.25307533733522518018502556836, 5.99478621779306996641276815298, 6.77585713292461089085449624996, 7.52549781412875013337099053754

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