Properties

Label 2-9702-1.1-c1-0-168
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 + 1.41·5-s + 8-s + 1.41·10-s + 11-s + 1.41·13-s + 16-s − 1.41·17-s − 2.82·19-s + 1.41·20-s + 22-s − 4·23-s − 2.99·25-s + 1.41·26-s − 6·29-s − 2.82·31-s + 32-s − 1.41·34-s − 8·37-s − 2.82·38-s + 1.41·40-s − 7.07·41-s − 8·43-s + 44-s − 4·46-s − 8.48·47-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 + 0.392·13-s + 0.250·16-s − 0.342·17-s − 0.648·19-s + 0.316·20-s + 0.213·22-s − 0.834·23-s − 0.599·25-s + 0.277·26-s − 1.11·29-s − 0.508·31-s + 0.176·32-s − 0.242·34-s − 1.31·37-s − 0.458·38-s + 0.223·40-s − 1.10·41-s − 1.21·43-s + 0.150·44-s − 0.589·46-s − 1.23·47-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: \(1\)
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 \( 1 - 1.41T + 5T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 - 9.89T + 89T^{2} \)
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

−7.19576533044290010539002298808, −6.40238640090810746421784041912, −6.05560860576785897999365049570, −5.27266830018561130165696431117, −4.62196015954846721481784902781, −3.74389898343624420507283980275, −3.24542672905485775262595585305, −1.94697167838855427000488868940, −1.74846006970111332048372191005, 0, 1.74846006970111332048372191005, 1.94697167838855427000488868940, 3.24542672905485775262595585305, 3.74389898343624420507283980275, 4.62196015954846721481784902781, 5.27266830018561130165696431117, 6.05560860576785897999365049570, 6.40238640090810746421784041912, 7.19576533044290010539002298808

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