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Properties

Label 1-1084-1084.1083-r0-0-0
Degree $1$
Conductor $1084$
Sign $1$
Analytic cond. $5.03407$
Root an. cond. $5.03407$
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  + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 35-s + 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s + 53-s − 55-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 35-s + 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s + 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1084 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1084 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1084\)    =    \(2^{2} \cdot 271\)
Sign: $1$
Analytic conductor: \(5.03407\)
Root analytic conductor: \(5.03407\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1084} (1083, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1084,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.364650989\)
\(L(\frac12)\) \(\approx\) \(2.364650989\)
\(L(1)\) \(\approx\) \(1.589701379\)
\(L(1)\) \(\approx\) \(1.589701379\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
271 \( 1 \)
good3 \( 1 + T \)
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

−21.36876739380718981526917196009, −20.64265591474535173204027257835, −20.01318848438752282350734357517, −19.00572062542435660180944621029, −18.59827304734206246978600844464, −17.66717354317984932193584394433, −16.60639334410313745267756875871, −16.064589853901833409300431908541, −14.97474394839996255151014069364, −14.4299593802416110533836841491, −13.49258108285272332226093601729, −12.96006400867102070969874466966, −12.36467337489183209720926460251, −10.843116452346302462125455363334, −9.90466232138504060506995852092, −9.59790847163553342026418887627, −8.819137255043499654301750356634, −7.47173851551309668699144612884, −7.21413483732980085131481362038, −5.80384246310714385400932623379, −5.19868669396547351834557240187, −3.86590828341466188915600756854, −2.79768435449995153014651920825, −2.47015095668009552276306539312, −1.08043766767532269781637326902, 1.08043766767532269781637326902, 2.47015095668009552276306539312, 2.79768435449995153014651920825, 3.86590828341466188915600756854, 5.19868669396547351834557240187, 5.80384246310714385400932623379, 7.21413483732980085131481362038, 7.47173851551309668699144612884, 8.819137255043499654301750356634, 9.59790847163553342026418887627, 9.90466232138504060506995852092, 10.843116452346302462125455363334, 12.36467337489183209720926460251, 12.96006400867102070969874466966, 13.49258108285272332226093601729, 14.4299593802416110533836841491, 14.97474394839996255151014069364, 16.064589853901833409300431908541, 16.60639334410313745267756875871, 17.66717354317984932193584394433, 18.59827304734206246978600844464, 19.00572062542435660180944621029, 20.01318848438752282350734357517, 20.64265591474535173204027257835, 21.36876739380718981526917196009

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