Properties

Label 2-483-1.1-c1-0-5
Degree $2$
Conductor $483$
Sign $1$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 3-s + 0.618·4-s + 1.38·5-s − 1.61·6-s + 7-s + 2.23·8-s + 9-s − 2.23·10-s + 11-s + 0.618·12-s − 1.61·13-s − 1.61·14-s + 1.38·15-s − 4.85·16-s + 3.47·17-s − 1.61·18-s − 0.236·19-s + 0.854·20-s + 21-s − 1.61·22-s + 23-s + 2.23·24-s − 3.09·25-s + 2.61·26-s + 27-s + 0.618·28-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.618·5-s − 0.660·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.707·10-s + 0.301·11-s + 0.178·12-s − 0.448·13-s − 0.432·14-s + 0.356·15-s − 1.21·16-s + 0.842·17-s − 0.381·18-s − 0.0541·19-s + 0.190·20-s + 0.218·21-s − 0.344·22-s + 0.208·23-s + 0.456·24-s − 0.618·25-s + 0.513·26-s + 0.192·27-s + 0.116·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.092389688\)
\(L(\frac12)\) \(\approx\) \(1.092389688\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 - T \)
good2 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 - 1.38T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 - 3.47T + 17T^{2} \)
19 \( 1 + 0.236T + 19T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 + 1.70T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 + 1.14T + 61T^{2} \)
67 \( 1 - 3.09T + 67T^{2} \)
71 \( 1 - 5.32T + 71T^{2} \)
73 \( 1 - 6.23T + 73T^{2} \)
79 \( 1 + 9.76T + 79T^{2} \)
83 \( 1 + 0.0557T + 83T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 + 6.70T + 97T^{2} \)
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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

−10.51956163342141614770249780371, −9.925266876211551147261266414720, −9.234324300939714846843309286488, −8.383307199888269603550199521902, −7.69029462585363045915401481963, −6.69895685576002503227698034939, −5.32511920719835292083451807746, −4.14945186551961343812603338772, −2.51826894006238550268856117590, −1.24614586394501466118696681400, 1.24614586394501466118696681400, 2.51826894006238550268856117590, 4.14945186551961343812603338772, 5.32511920719835292083451807746, 6.69895685576002503227698034939, 7.69029462585363045915401481963, 8.383307199888269603550199521902, 9.234324300939714846843309286488, 9.925266876211551147261266414720, 10.51956163342141614770249780371

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