↑

Properties

Label 2-483-161.10-c1-0-15
Degree $2$
Conductor $483$
Sign $-0.0170 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.340 + 0.985i)2-s + (0.458 + 0.888i)3-s + (0.718 + 0.564i)4-s + (3.21 − 0.306i)5-s + (−1.03 + 0.148i)6-s + (−1.39 − 2.25i)7-s + (−2.55 + 1.64i)8-s + (−0.580 + 0.814i)9-s + (−0.792 + 3.26i)10-s + (3.42 − 1.18i)11-s + (−0.172 + 0.896i)12-s + (1.57 + 5.35i)13-s + (2.69 − 0.603i)14-s + (1.74 + 2.71i)15-s + (−0.315 − 1.30i)16-s + (−5.14 − 2.05i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.696i)2-s + (0.264 + 0.513i)3-s + (0.359 + 0.282i)4-s + (1.43 − 0.137i)5-s + (−0.421 + 0.0605i)6-s + (−0.525 − 0.850i)7-s + (−0.903 + 0.580i)8-s + (−0.193 + 0.271i)9-s + (−0.250 + 1.03i)10-s + (1.03 − 0.357i)11-s + (−0.0499 + 0.258i)12-s + (0.435 + 1.48i)13-s + (0.719 − 0.161i)14-s + (0.450 + 0.700i)15-s + (−0.0788 − 0.325i)16-s + (−1.24 − 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0170 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0170 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.0170 - 0.999i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.0170 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28512 + 1.30722i\)
\(L(\frac12)\) \(\approx\) \(1.28512 + 1.30722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (1.39 + 2.25i)T \)
23 \( 1 + (0.334 - 4.78i)T \)
good2 \( 1 + (0.340 - 0.985i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-3.21 + 0.306i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.42 + 1.18i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-1.57 - 5.35i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (5.14 + 2.05i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-3.70 + 1.48i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.591 + 4.11i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-6.14 + 0.292i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (3.45 + 2.46i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (5.01 + 2.29i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.31 + 5.15i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (9.04 - 5.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.89 - 3.03i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (2.41 + 0.586i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (7.25 + 3.74i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.76 - 9.14i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (4.30 + 4.97i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.956 + 1.21i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-6.81 + 7.14i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (6.29 + 13.7i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.436 + 9.17i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-5.18 + 11.3i)T + (-63.5 - 73.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.21916475505893621894888088594, −10.00076183125101851541760506287, −9.204149810329879158252874960922, −8.876610654635263727635688040669, −7.34516037777196010953312048731, −6.56085529302061378512598995701, −5.97361529224381313555053756926, −4.53249796791002570903566975367, −3.30032789359200762723608350570, −1.87453410005319789012971011694, 1.37195161334207460111571958117, 2.37656117366220652986300893881, 3.25556358600645400662918082513, 5.35431227876373572810679205725, 6.37019923364096957158942073536, 6.58303838928446840151193024306, 8.387879341352658803362789565486, 9.204494391212842439364616881614, 9.913545058808646193953151995797, 10.57616229860988066350813238628

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