Properties

Label 2-1003-1.1-c1-0-10
Degree $2$
Conductor $1003$
Sign $1$
Analytic cond. $8.00899$
Root an. cond. $2.83001$
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  − 1.84·2-s − 2.22·3-s + 1.41·4-s − 0.984·5-s + 4.11·6-s + 2.90·7-s + 1.08·8-s + 1.97·9-s + 1.81·10-s + 3.12·11-s − 3.14·12-s − 2.44·13-s − 5.37·14-s + 2.19·15-s − 4.83·16-s + 17-s − 3.64·18-s + 5.79·19-s − 1.39·20-s − 6.48·21-s − 5.77·22-s + 3.79·23-s − 2.42·24-s − 4.03·25-s + 4.50·26-s + 2.29·27-s + 4.10·28-s + ⋯
L(s)  = 1  − 1.30·2-s − 1.28·3-s + 0.705·4-s − 0.440·5-s + 1.68·6-s + 1.09·7-s + 0.384·8-s + 0.656·9-s + 0.575·10-s + 0.942·11-s − 0.908·12-s − 0.676·13-s − 1.43·14-s + 0.566·15-s − 1.20·16-s + 0.242·17-s − 0.858·18-s + 1.32·19-s − 0.310·20-s − 1.41·21-s − 1.23·22-s + 0.791·23-s − 0.494·24-s − 0.806·25-s + 0.884·26-s + 0.441·27-s + 0.775·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $1$
Analytic conductor: \(8.00899\)
Root analytic conductor: \(2.83001\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4720935970\)
\(L(\frac12)\) \(\approx\) \(0.4720935970\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
59 \( 1 + T \)
good2 \( 1 + 1.84T + 2T^{2} \)
3 \( 1 + 2.22T + 3T^{2} \)
5 \( 1 + 0.984T + 5T^{2} \)
7 \( 1 - 2.90T + 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
19 \( 1 - 5.79T + 19T^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 + 4.79T + 29T^{2} \)
31 \( 1 + 9.81T + 31T^{2} \)
37 \( 1 + 5.03T + 37T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 - 6.32T + 47T^{2} \)
53 \( 1 - 8.94T + 53T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 2.57T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 5.47T + 83T^{2} \)
89 \( 1 - 6.79T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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

−9.935424209703596704347304275951, −9.212526971445790030668996102710, −8.361393320423935180890147136539, −7.38196530623885303969489018067, −7.03511813915147844957744009748, −5.58075540239079898273293949068, −4.98886181822218094093475999180, −3.84389836516423722903416740662, −1.81051413299496618952460941267, −0.72164581606979712081702071594, 0.72164581606979712081702071594, 1.81051413299496618952460941267, 3.84389836516423722903416740662, 4.98886181822218094093475999180, 5.58075540239079898273293949068, 7.03511813915147844957744009748, 7.38196530623885303969489018067, 8.361393320423935180890147136539, 9.212526971445790030668996102710, 9.935424209703596704347304275951

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