Properties

Label 2-1002-1.1-c1-0-25
Degree $2$
Conductor $1002$
Sign $-1$
Analytic cond. $8.00101$
Root an. cond. $2.82860$
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-s + 3-s + 4-s + 2·5-s − 6-s − 4·7-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 4·14-s + 2·15-s + 16-s − 4·17-s − 18-s − 4·19-s + 2·20-s − 4·21-s + 4·22-s − 4·23-s − 24-s − 25-s + 27-s − 4·28-s − 2·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s − 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1002\)    =    \(2 \cdot 3 \cdot 167\)
Sign: $-1$
Analytic conductor: \(8.00101\)
Root analytic conductor: \(2.82860\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1002} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1002,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.562384110123212066972771572541, −8.872167065392671989876307770954, −8.076909988757103929513574005725, −6.99930492560334263539425557990, −6.34388139091273497511950452640, −5.47307925839941586446907478099, −3.95788562114956825460286935708, −2.75024761094662395043917722046, −2.08141000487309969375535467484, 0, 2.08141000487309969375535467484, 2.75024761094662395043917722046, 3.95788562114956825460286935708, 5.47307925839941586446907478099, 6.34388139091273497511950452640, 6.99930492560334263539425557990, 8.076909988757103929513574005725, 8.872167065392671989876307770954, 9.562384110123212066972771572541

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