Properties

Label 2-9702-1.1-c1-0-68
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.473·5-s − 8-s − 0.473·10-s − 11-s + 7.03·13-s + 16-s + 4.64·17-s + 6.55·19-s + 0.473·20-s + 22-s − 1.49·23-s − 4.77·25-s − 7.03·26-s + 2.11·29-s + 6.83·31-s − 32-s − 4.64·34-s − 4.32·37-s − 6.55·38-s − 0.473·40-s + 11.2·41-s + 0.158·43-s − 44-s + 1.49·46-s − 0.270·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.211·5-s − 0.353·8-s − 0.149·10-s − 0.301·11-s + 1.95·13-s + 0.250·16-s + 1.12·17-s + 1.50·19-s + 0.105·20-s + 0.213·22-s − 0.312·23-s − 0.955·25-s − 1.37·26-s + 0.393·29-s + 1.22·31-s − 0.176·32-s − 0.796·34-s − 0.711·37-s − 1.06·38-s − 0.0748·40-s + 1.75·41-s + 0.0241·43-s − 0.150·44-s + 0.220·46-s − 0.0394·47-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

\(L(1)\) \(\approx\) \(2.020370371\)
\(L(\frac12)\) \(\approx\) \(2.020370371\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.473T + 5T^{2} \)
13 \( 1 - 7.03T + 13T^{2} \)
17 \( 1 - 4.64T + 17T^{2} \)
19 \( 1 - 6.55T + 19T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 - 2.11T + 29T^{2} \)
31 \( 1 - 6.83T + 31T^{2} \)
37 \( 1 + 4.32T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 0.158T + 43T^{2} \)
47 \( 1 + 0.270T + 47T^{2} \)
53 \( 1 - 9.66T + 53T^{2} \)
59 \( 1 + 9.82T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 1.21T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + 6.13T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 7.54T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.73146662358026580679163714062, −7.22135846105435075861225490138, −6.12823078120035590030121501004, −5.93250233921562734951934839974, −5.11771542428304673747607150832, −3.97705405554895215965815780523, −3.36553942625143503710044523424, −2.55373992306927247605057385042, −1.40925759376384295005214542228, −0.856578081408337407676950322815, 0.856578081408337407676950322815, 1.40925759376384295005214542228, 2.55373992306927247605057385042, 3.36553942625143503710044523424, 3.97705405554895215965815780523, 5.11771542428304673747607150832, 5.93250233921562734951934839974, 6.12823078120035590030121501004, 7.22135846105435075861225490138, 7.73146662358026580679163714062

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