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Properties

Label 1-56-56.27-r0-0-0
Degree $1$
Conductor $56$
Sign $1$
Analytic cond. $0.260062$
Root an. cond. $0.260062$
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 + 9-s + 11-s + 13-s − 15-s − 17-s − 19-s − 23-s + 25-s − 27-s − 29-s + 31-s − 33-s − 37-s − 39-s − 41-s + 43-s + 45-s + 47-s + 51-s − 53-s + 55-s + 57-s − 59-s + 61-s + 65-s + ⋯
L(s)  = 1  − 3-s + 5-s + 9-s + 11-s + 13-s − 15-s − 17-s − 19-s − 23-s + 25-s − 27-s − 29-s + 31-s − 33-s − 37-s − 39-s − 41-s + 43-s + 45-s + 47-s + 51-s − 53-s + 55-s + 57-s − 59-s + 61-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.260062\)
Root analytic conductor: \(0.260062\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56} (27, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 56,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7970666386\)
\(L(\frac12)\) \(\approx\) \(0.7970666386\)
\(L(1)\) \(\approx\) \(0.9087107831\)
\(L(1)\) \(\approx\) \(0.9087107831\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + T \)
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

−33.21384508413441556947981994086, −32.26805299301437722833074587065, −30.37885996365555324724710854790, −29.663123097826437709287460597343, −28.50178767498019244594036118876, −27.73009191200168968338538357462, −26.233969008053320449388405517235, −24.99179580441211699441168341252, −23.928056823052222359144165414751, −22.563981192226182582279563216899, −21.80816437561717857683283978021, −20.60943721127255231322179520360, −18.8687446212381257790910041455, −17.679014816658470466689971178564, −16.96087601038398759411881077654, −15.61213364330574071489733955479, −13.943650781163097639101385420720, −12.77810642984479434694729706504, −11.37581416061934331022277656854, −10.254433791373370891588126331653, −8.88129936992132932367060551338, −6.68317342370464988339037119070, −5.86956152172763602899718494129, −4.228883255203880686042267075302, −1.70166613709380085467512386443, 1.70166613709380085467512386443, 4.228883255203880686042267075302, 5.86956152172763602899718494129, 6.68317342370464988339037119070, 8.88129936992132932367060551338, 10.254433791373370891588126331653, 11.37581416061934331022277656854, 12.77810642984479434694729706504, 13.943650781163097639101385420720, 15.61213364330574071489733955479, 16.96087601038398759411881077654, 17.679014816658470466689971178564, 18.8687446212381257790910041455, 20.60943721127255231322179520360, 21.80816437561717857683283978021, 22.563981192226182582279563216899, 23.928056823052222359144165414751, 24.99179580441211699441168341252, 26.233969008053320449388405517235, 27.73009191200168968338538357462, 28.50178767498019244594036118876, 29.663123097826437709287460597343, 30.37885996365555324724710854790, 32.26805299301437722833074587065, 33.21384508413441556947981994086

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