Properties

Label 2-983-983.982-c0-0-7
Degree $2$
Conductor $983$
Sign $1$
Analytic cond. $0.490580$
Root an. cond. $0.700414$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.116·2-s + 1.19·3-s − 0.986·4-s − 0.138·6-s + 0.792·7-s + 0.231·8-s + 0.426·9-s − 1.17·12-s − 0.0921·14-s + 0.959·16-s − 0.0495·18-s + 1.78·19-s + 0.946·21-s − 1.37·23-s + 0.275·24-s + 25-s − 0.685·27-s − 0.781·28-s + 0.347·31-s − 0.342·32-s − 0.420·36-s − 1.87·37-s − 0.207·38-s + 1.53·41-s − 0.110·42-s − 1.67·43-s + 0.159·46-s + ⋯
L(s)  = 1  − 0.116·2-s + 1.19·3-s − 0.986·4-s − 0.138·6-s + 0.792·7-s + 0.231·8-s + 0.426·9-s − 1.17·12-s − 0.0921·14-s + 0.959·16-s − 0.0495·18-s + 1.78·19-s + 0.946·21-s − 1.37·23-s + 0.275·24-s + 25-s − 0.685·27-s − 0.781·28-s + 0.347·31-s − 0.342·32-s − 0.420·36-s − 1.87·37-s − 0.207·38-s + 1.53·41-s − 0.110·42-s − 1.67·43-s + 0.159·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(983\)
Sign: $1$
Analytic conductor: \(0.490580\)
Root analytic conductor: \(0.700414\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{983} (982, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 983,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227170521\)
\(L(\frac12)\) \(\approx\) \(1.227170521\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad983 \( 1 - T \)
good2 \( 1 + 0.116T + T^{2} \)
3 \( 1 - 1.19T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - 0.792T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.78T + T^{2} \)
23 \( 1 + 1.37T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.347T + T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 - 1.53T + T^{2} \)
43 \( 1 + 1.67T + T^{2} \)
47 \( 1 - 1.94T + T^{2} \)
53 \( 1 + 1.67T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.53T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.98T + T^{2} \)
97 \( 1 - T^{2} \)
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

−9.843493989929148960598492976864, −9.311648883950822854084801764072, −8.454170294418607788866434192844, −8.027832134127019535465875535663, −7.21361135051369251398909862127, −5.67609280776698598832633927088, −4.85015688062753448610314471449, −3.84405139581806485856280208519, −2.95587660680363076836170497980, −1.53106581503334650470559092247, 1.53106581503334650470559092247, 2.95587660680363076836170497980, 3.84405139581806485856280208519, 4.85015688062753448610314471449, 5.67609280776698598832633927088, 7.21361135051369251398909862127, 8.027832134127019535465875535663, 8.454170294418607788866434192844, 9.311648883950822854084801764072, 9.843493989929148960598492976864

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