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Properties

Label 1-1013-1013.1012-r0-0-0
Degree $1$
Conductor $1013$
Sign $1$
Analytic cond. $4.70435$
Root an. cond. $4.70435$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1013\)
Sign: $1$
Analytic conductor: \(4.70435\)
Root analytic conductor: \(4.70435\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1013} (1012, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1013,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4279146528\)
\(L(\frac12)\) \(\approx\) \(0.4279146528\)
\(L(1)\) \(\approx\) \(0.4290375233\)
\(L(1)\) \(\approx\) \(0.4290375233\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1013 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.71013509891824159994121549216, −20.546727842136064607310763249, −19.89652290619082733599250931158, −19.11600110826625612162349408927, −18.536757497188753844101698322290, −17.76534817309231194380028241922, −16.720860022197530990505683766027, −16.409161642340478920723313730, −15.59654275579086545593411016586, −15.081274657105066165937692527434, −13.47417713990504027421814690919, −12.51172679907626270733757152774, −11.83557890960561129679187773443, −11.10819184889596471454704409012, −10.617539060379001899023475022881, −9.33745725015286866263410468933, −8.96641817212610903667197949948, −7.647795427684475806187134151392, −6.86019300420956206350674734107, −6.41167061668114460215750157763, −5.31099077184320002150698348429, −3.90550743947475962169522577397, −3.28454039928955363641087523672, −1.613687908800275422036005715113, −0.59358790904296108818752068785, 0.59358790904296108818752068785, 1.613687908800275422036005715113, 3.28454039928955363641087523672, 3.90550743947475962169522577397, 5.31099077184320002150698348429, 6.41167061668114460215750157763, 6.86019300420956206350674734107, 7.647795427684475806187134151392, 8.96641817212610903667197949948, 9.33745725015286866263410468933, 10.617539060379001899023475022881, 11.10819184889596471454704409012, 11.83557890960561129679187773443, 12.51172679907626270733757152774, 13.47417713990504027421814690919, 15.081274657105066165937692527434, 15.59654275579086545593411016586, 16.409161642340478920723313730, 16.720860022197530990505683766027, 17.76534817309231194380028241922, 18.536757497188753844101698322290, 19.11600110826625612162349408927, 19.89652290619082733599250931158, 20.546727842136064607310763249, 21.71013509891824159994121549216

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