Properties

Label 2-1003-1.1-c1-0-64
Degree $2$
Conductor $1003$
Sign $-1$
Analytic cond. $8.00899$
Root an. cond. $2.83001$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s − 2·5-s + 2·7-s + 9-s − 5·11-s − 4·12-s + 2·13-s − 4·15-s + 4·16-s − 17-s − 5·19-s + 4·20-s + 4·21-s − 5·23-s − 25-s − 4·27-s − 4·28-s − 6·29-s + 8·31-s − 10·33-s − 4·35-s − 2·36-s + 2·37-s + 4·39-s − 6·41-s − 2·43-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 1.15·12-s + 0.554·13-s − 1.03·15-s + 16-s − 0.242·17-s − 1.14·19-s + 0.894·20-s + 0.872·21-s − 1.04·23-s − 1/5·25-s − 0.769·27-s − 0.755·28-s − 1.11·29-s + 1.43·31-s − 1.74·33-s − 0.676·35-s − 1/3·36-s + 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.350338958223820564660586341669, −8.392005801640334209926396341161, −8.136466910523484092759311461600, −7.63852837306894457822902912923, −6.01428375027425316853400492481, −4.88154560188503289286554894395, −4.13215293550911202611023429139, −3.27629583709466000908159136571, −2.05439478221516503924411366935, 0, 2.05439478221516503924411366935, 3.27629583709466000908159136571, 4.13215293550911202611023429139, 4.88154560188503289286554894395, 6.01428375027425316853400492481, 7.63852837306894457822902912923, 8.136466910523484092759311461600, 8.392005801640334209926396341161, 9.350338958223820564660586341669

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