Properties

Label 2-9702-1.1-c1-0-127
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 $1$

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 − 4·13-s + 16-s + 6·17-s − 2·19-s + 2·20-s + 22-s − 25-s + 4·26-s + 6·29-s − 2·31-s − 32-s − 6·34-s + 2·37-s + 2·38-s − 2·40-s − 2·41-s − 12·43-s − 44-s − 6·47-s + 50-s − 4·52-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 − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.784·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.324·38-s − 0.316·40-s − 0.312·41-s − 1.82·43-s − 0.150·44-s − 0.875·47-s + 0.141·50-s − 0.554·52-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−7.46758067332539910168084752489, −6.64844043492678762667548571037, −6.13947703686177335991368807841, −5.26964445931964368989458250501, −4.85404470743878570796267057581, −3.59335919373833731278220557054, −2.81163540902542034522408294565, −2.06347451200798813330821105283, −1.26397067800308264554359809769, 0, 1.26397067800308264554359809769, 2.06347451200798813330821105283, 2.81163540902542034522408294565, 3.59335919373833731278220557054, 4.85404470743878570796267057581, 5.26964445931964368989458250501, 6.13947703686177335991368807841, 6.64844043492678762667548571037, 7.46758067332539910168084752489

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