Properties

Label 2-9702-1.1-c1-0-165
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 + 1.64·5-s + 8-s + 1.64·10-s − 11-s + 5·13-s + 16-s − 6·17-s − 5.64·19-s + 1.64·20-s − 22-s − 1.64·23-s − 2.29·25-s + 5·26-s − 6.29·29-s − 4·31-s + 32-s − 6·34-s + 3.64·37-s − 5.64·38-s + 1.64·40-s − 10.9·41-s − 4·43-s − 44-s − 1.64·46-s − 2.70·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.736·5-s + 0.353·8-s + 0.520·10-s − 0.301·11-s + 1.38·13-s + 0.250·16-s − 1.45·17-s − 1.29·19-s + 0.368·20-s − 0.213·22-s − 0.343·23-s − 0.458·25-s + 0.980·26-s − 1.16·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.599·37-s − 0.915·38-s + 0.260·40-s − 1.70·41-s − 0.609·43-s − 0.150·44-s − 0.242·46-s − 0.395·47-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 \( 1 - 1.64T + 5T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 + 6.29T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 3.64T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 + 1.64T + 53T^{2} \)
59 \( 1 + 4.64T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + 4.35T + 71T^{2} \)
73 \( 1 - 0.354T + 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + 6.58T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−7.04090861367201616107273083799, −6.52326703744731552005784216862, −5.92775365997783703085263994456, −5.42068830990256353739443951677, −4.45312145039469225057659130503, −3.94222302563073391650735513427, −3.09881866036184492050100659445, −2.04818959901523960769509085253, −1.69174668692843826766716863750, 0, 1.69174668692843826766716863750, 2.04818959901523960769509085253, 3.09881866036184492050100659445, 3.94222302563073391650735513427, 4.45312145039469225057659130503, 5.42068830990256353739443951677, 5.92775365997783703085263994456, 6.52326703744731552005784216862, 7.04090861367201616107273083799

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