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Properties

Label 1-552-552.413-r0-0-0
Degree $1$
Conductor $552$
Sign $1$
Analytic cond. $2.56347$
Root an. cond. $2.56347$
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  − 5-s − 7-s − 11-s − 13-s + 17-s + 19-s + 25-s + 29-s + 31-s + 35-s + 37-s − 41-s + 43-s − 47-s + 49-s − 53-s + 55-s + 59-s + 61-s + 65-s + 67-s − 71-s + 73-s + 77-s − 79-s − 83-s − 85-s + ⋯
L(s)  = 1  − 5-s − 7-s − 11-s − 13-s + 17-s + 19-s + 25-s + 29-s + 31-s + 35-s + 37-s − 41-s + 43-s − 47-s + 49-s − 53-s + 55-s + 59-s + 61-s + 65-s + 67-s − 71-s + 73-s + 77-s − 79-s − 83-s − 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{552} (413, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8239815083\)
\(L(\frac12)\) \(\approx\) \(0.8239815083\)
\(L(1)\) \(\approx\) \(0.7734825997\)
\(L(1)\) \(\approx\) \(0.7734825997\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
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

−23.28506110378513251415142794634, −22.66905007750410205768022480712, −21.80196336631549197468485222477, −20.73358809225374352590295226729, −19.90499893086991871501820983272, −19.18047124040358235957680606975, −18.57686436884048915689914444649, −17.435442070330085374890941334438, −16.30454491646441678573218335670, −15.91218737306652937810926721112, −15.01051099658937445007417074469, −14.02267759483199644717686100826, −12.90913828033179174829104845706, −12.26115969397875301012145653621, −11.45815011400856301239637819721, −10.19917070182665691152070380383, −9.69282508318951239625832375842, −8.31530848455138401074346175571, −7.594368389265276250834934799018, −6.75795062546088844896004350885, −5.493037518391693817845890490846, −4.56255666773272522965007177979, −3.313821073631981541574969973974, −2.694065124211943818387918595920, −0.72449390338228455109538531627, 0.72449390338228455109538531627, 2.694065124211943818387918595920, 3.313821073631981541574969973974, 4.56255666773272522965007177979, 5.493037518391693817845890490846, 6.75795062546088844896004350885, 7.594368389265276250834934799018, 8.31530848455138401074346175571, 9.69282508318951239625832375842, 10.19917070182665691152070380383, 11.45815011400856301239637819721, 12.26115969397875301012145653621, 12.90913828033179174829104845706, 14.02267759483199644717686100826, 15.01051099658937445007417074469, 15.91218737306652937810926721112, 16.30454491646441678573218335670, 17.435442070330085374890941334438, 18.57686436884048915689914444649, 19.18047124040358235957680606975, 19.90499893086991871501820983272, 20.73358809225374352590295226729, 21.80196336631549197468485222477, 22.66905007750410205768022480712, 23.28506110378513251415142794634

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