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Properties

Label 2-483-21.20-c1-0-34
Degree $2$
Conductor $483$
Sign $-0.0799 + 0.996i$
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  − 2.33i·2-s + (1.20 + 1.24i)3-s − 3.46·4-s + 1.30·5-s + (2.90 − 2.81i)6-s + (1.98 − 1.74i)7-s + 3.42i·8-s + (−0.0963 + 2.99i)9-s − 3.05i·10-s + 3.74i·11-s + (−4.17 − 4.31i)12-s − 4.46i·13-s + (−4.08 − 4.64i)14-s + (1.57 + 1.62i)15-s + 1.07·16-s + 7.95·17-s + ⋯
L(s)  = 1  − 1.65i·2-s + (0.695 + 0.718i)3-s − 1.73·4-s + 0.585·5-s + (1.18 − 1.14i)6-s + (0.750 − 0.660i)7-s + 1.20i·8-s + (−0.0321 + 0.999i)9-s − 0.967i·10-s + 1.12i·11-s + (−1.20 − 1.24i)12-s − 1.23i·13-s + (−1.09 − 1.24i)14-s + (0.407 + 0.420i)15-s + 0.267·16-s + 1.92·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0799 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0799 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.0799 + 0.996i$
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: \((2,\ 483,\ (\ :1/2),\ -0.0799 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31697 - 1.42686i\)
\(L(\frac12)\) \(\approx\) \(1.31697 - 1.42686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.20 - 1.24i)T \)
7 \( 1 + (-1.98 + 1.74i)T \)
23 \( 1 + iT \)
good2 \( 1 + 2.33iT - 2T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
11 \( 1 - 3.74iT - 11T^{2} \)
13 \( 1 + 4.46iT - 13T^{2} \)
17 \( 1 - 7.95T + 17T^{2} \)
19 \( 1 - 0.403iT - 19T^{2} \)
29 \( 1 + 5.43iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 - 3.84T + 37T^{2} \)
41 \( 1 + 0.835T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + 2.04T + 47T^{2} \)
53 \( 1 - 13.4iT - 53T^{2} \)
59 \( 1 + 1.03T + 59T^{2} \)
61 \( 1 + 1.07iT - 61T^{2} \)
67 \( 1 + 8.02T + 67T^{2} \)
71 \( 1 - 4.80iT - 71T^{2} \)
73 \( 1 - 16.3iT - 73T^{2} \)
79 \( 1 - 2.39T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 1.95iT - 97T^{2} \)
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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

−10.45424786910394860518062733450, −10.00978499958115339884455960603, −9.614867457508786409192475272656, −8.247424105985159998445697145151, −7.57839409354890862412127405938, −5.50405203128525293606346832971, −4.56500354939948814118428969821, −3.63526055152110707655658934551, −2.57675691618820774764396688667, −1.42869410506164153998982532668, 1.68275717931646090442466807120, 3.39409697515673359665190972385, 5.04445236883697268386304747714, 5.87366174614319817454468548246, 6.60614972389275773451184775100, 7.64106345507464006054706189889, 8.311110741186316567408342369031, 8.963558854689815846667443977026, 9.770848829578029286227117016561, 11.43657566376863728533250972238

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