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Properties

Label 2-483-21.20-c1-0-9
Degree $2$
Conductor $483$
Sign $-0.370 - 0.928i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.382 − 1.68i)3-s + 4-s − 2.93·5-s + (1.68 − 0.382i)6-s + (0.414 + 2.61i)7-s + 3i·8-s + (−2.70 + 1.29i)9-s − 2.93i·10-s + 0.585i·11-s + (−0.382 − 1.68i)12-s + 1.53i·13-s + (−2.61 + 0.414i)14-s + (1.12 + 4.94i)15-s − 16-s − 0.317·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.220 − 0.975i)3-s + 0.5·4-s − 1.31·5-s + (0.689 − 0.156i)6-s + (0.156 + 0.987i)7-s + 1.06i·8-s + (−0.902 + 0.430i)9-s − 0.926i·10-s + 0.176i·11-s + (−0.110 − 0.487i)12-s + 0.424i·13-s + (−0.698 + 0.110i)14-s + (0.289 + 1.27i)15-s − 0.250·16-s − 0.0768·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.370 - 0.928i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531591 + 0.784737i\)
\(L(\frac12)\) \(\approx\) \(0.531591 + 0.784737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 + 1.68i)T \)
7 \( 1 + (-0.414 - 2.61i)T \)
23 \( 1 - iT \)
good2 \( 1 - iT - 2T^{2} \)
5 \( 1 + 2.93T + 5T^{2} \)
11 \( 1 - 0.585iT - 11T^{2} \)
13 \( 1 - 1.53iT - 13T^{2} \)
17 \( 1 + 0.317T + 17T^{2} \)
19 \( 1 - 2.16iT - 19T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 6.62iT - 31T^{2} \)
37 \( 1 + 6.48T + 37T^{2} \)
41 \( 1 - 9.55T + 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 - 4.01T + 47T^{2} \)
53 \( 1 + 4.82iT - 53T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 9.65iT - 71T^{2} \)
73 \( 1 - 5.86iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 8.92T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 2.74iT - 97T^{2} \)
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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

−11.42634932604321715895046996519, −10.79933466050089529084066471813, −8.974465985773761464163657751274, −8.236684694990239949379211193446, −7.54176830104565556380360937783, −6.81852449644786189527142264241, −5.89164053519587205144984168495, −4.91291058986125806844661571999, −3.19478368201143597473481859625, −1.86549875571546383250136321346, 0.57728473446826275768635673281, 2.84584193207384649665915629328, 3.89260520913756101580820335646, 4.39953434026058700569916691185, 5.95638664812005055473360102225, 7.20929264348804694956601299999, 7.930249808281338175349929302524, 9.149639105261210960588117563466, 10.22607910060727493467395640573, 10.78851146329733677418863865762

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