Properties

Label 2-9702-1.1-c1-0-65
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 + 0.414·5-s + 8-s + 0.414·10-s + 11-s + 1.17·13-s + 16-s + 2.17·17-s − 0.828·19-s + 0.414·20-s + 22-s − 3.24·23-s − 4.82·25-s + 1.17·26-s + 2.82·29-s + 6.48·31-s + 32-s + 2.17·34-s + 9.65·37-s − 0.828·38-s + 0.414·40-s − 4.65·41-s − 2.82·43-s + 44-s − 3.24·46-s + 9.24·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.185·5-s + 0.353·8-s + 0.130·10-s + 0.301·11-s + 0.324·13-s + 0.250·16-s + 0.526·17-s − 0.190·19-s + 0.0926·20-s + 0.213·22-s − 0.676·23-s − 0.965·25-s + 0.229·26-s + 0.525·29-s + 1.16·31-s + 0.176·32-s + 0.372·34-s + 1.58·37-s − 0.134·38-s + 0.0654·40-s − 0.727·41-s − 0.431·43-s + 0.150·44-s − 0.478·46-s + 1.34·47-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: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.791015634\)
\(L(\frac12)\) \(\approx\) \(3.791015634\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414T + 5T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 6.48T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 - 9.24T + 47T^{2} \)
53 \( 1 + 5.17T + 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 - 1.58T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 - 4.75T + 79T^{2} \)
83 \( 1 - 9.82T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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

−7.74508030871712014958543519534, −6.77625810119612415077633585751, −6.21059010224548089020215803050, −5.72383782792806327150762259784, −4.86021416685617960625481715593, −4.19459137599238394008301547240, −3.53573991178784864703653255170, −2.67916067054928753832559166848, −1.87934919272965384754664742806, −0.847669397327943641882464577860, 0.847669397327943641882464577860, 1.87934919272965384754664742806, 2.67916067054928753832559166848, 3.53573991178784864703653255170, 4.19459137599238394008301547240, 4.86021416685617960625481715593, 5.72383782792806327150762259784, 6.21059010224548089020215803050, 6.77625810119612415077633585751, 7.74508030871712014958543519534

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