Properties

Degree 2
Conductor 2003
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.347·3-s + 4-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 16-s − 19-s + 25-s − 0.652·27-s − 0.879·36-s + 0.532·39-s + 1.53·47-s + 0.347·48-s + 49-s + 1.53·52-s − 53-s − 0.347·57-s − 1.87·59-s + 64-s − 1.87·73-s + 0.347·75-s − 76-s − 1.87·79-s + 0.652·81-s − 89-s + 100-s + 1.53·101-s + ⋯
L(s)  = 1  + 0.347·3-s + 4-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 16-s − 19-s + 25-s − 0.652·27-s − 0.879·36-s + 0.532·39-s + 1.53·47-s + 0.347·48-s + 49-s + 1.53·52-s − 53-s − 0.347·57-s − 1.87·59-s + 64-s − 1.87·73-s + 0.347·75-s − 76-s − 1.87·79-s + 0.652·81-s − 89-s + 100-s + 1.53·101-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2003\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2003} (2002, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2003,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.629944499$
$L(\frac12)$  $\approx$  $1.629944499$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2003$, \(F_p\) is a polynomial of degree 2. If $p = 2003$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2003 \( 1+O(T) \)
good2 \( 1 - T^{2} \)
3 \( 1 - 0.347T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.53T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.53T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + 1.87T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.87T + T^{2} \)
79 \( 1 + 1.87T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.064888061779371384429558299190, −8.607462817657275460934427201340, −7.85305608860504719413333730021, −6.95700635433192104564140022369, −6.13648691027400823583051673066, −5.68031559858507047814753067409, −4.30350435254199879302278573543, −3.30057283543544462659792325523, −2.58262598939111800571373723922, −1.43042549484316165441871962923, 1.43042549484316165441871962923, 2.58262598939111800571373723922, 3.30057283543544462659792325523, 4.30350435254199879302278573543, 5.68031559858507047814753067409, 6.13648691027400823583051673066, 6.95700635433192104564140022369, 7.85305608860504719413333730021, 8.607462817657275460934427201340, 9.064888061779371384429558299190

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