Properties

Label 2-483-161.20-c1-0-6
Degree $2$
Conductor $483$
Sign $-0.831 + 0.554i$
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  + (0.308 + 2.14i)2-s + (0.909 + 0.415i)3-s + (−2.59 + 0.761i)4-s + (−1.43 + 1.66i)5-s + (−0.610 + 2.08i)6-s + (−1.37 − 2.25i)7-s + (−0.631 − 1.38i)8-s + (0.654 + 0.755i)9-s + (−4.00 − 2.57i)10-s + (−0.255 − 0.0367i)11-s + (−2.67 − 0.384i)12-s + (−3.68 + 5.73i)13-s + (4.42 − 3.65i)14-s + (−1.99 + 0.912i)15-s + (−1.77 + 1.13i)16-s + (−2.19 − 0.643i)17-s + ⋯
L(s)  = 1  + (0.218 + 1.51i)2-s + (0.525 + 0.239i)3-s + (−1.29 + 0.380i)4-s + (−0.643 + 0.742i)5-s + (−0.249 + 0.849i)6-s + (−0.520 − 0.853i)7-s + (−0.223 − 0.489i)8-s + (0.218 + 0.251i)9-s + (−1.26 − 0.814i)10-s + (−0.0771 − 0.0110i)11-s + (−0.771 − 0.110i)12-s + (−1.02 + 1.59i)13-s + (1.18 − 0.976i)14-s + (−0.515 + 0.235i)15-s + (−0.442 + 0.284i)16-s + (−0.531 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.831 + 0.554i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.831 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343362 - 1.13359i\)
\(L(\frac12)\) \(\approx\) \(0.343362 - 1.13359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (1.37 + 2.25i)T \)
23 \( 1 + (-3.53 + 3.23i)T \)
good2 \( 1 + (-0.308 - 2.14i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (1.43 - 1.66i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (0.255 + 0.0367i)T + (10.5 + 3.09i)T^{2} \)
13 \( 1 + (3.68 - 5.73i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (2.19 + 0.643i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (3.39 - 0.997i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-9.53 - 2.79i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-6.00 + 2.74i)T + (20.3 - 23.4i)T^{2} \)
37 \( 1 + (0.438 - 0.379i)T + (5.26 - 36.6i)T^{2} \)
41 \( 1 + (-7.48 - 6.48i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (5.35 + 2.44i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 - 3.99iT - 47T^{2} \)
53 \( 1 + (-3.89 - 6.05i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (-0.106 + 0.165i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-1.60 - 3.52i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-10.7 + 1.54i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (1.25 + 8.72i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (1.68 + 5.74i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (5.82 - 9.05i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (2.69 + 3.11i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (4.46 - 9.77i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (3.35 - 3.87i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.40627562614493171082056906244, −10.49593726575087370226451314375, −9.474063583840129816461839427462, −8.542937770473492977640248841329, −7.62559699869383872957053364108, −6.81704582871574545316161638244, −6.52400059257815989509871471143, −4.66961980892625747515544656822, −4.21009583952989177856229717841, −2.71425133783062940100784231109, 0.61904807064536688904160537116, 2.40400643839602114088309802203, 3.08605082927232167469532328120, 4.34391833534847989030851761315, 5.27040335917884473712792661336, 6.82454399237194560586079647630, 8.180795035743034396462018395806, 8.750698579174921563013746100449, 9.799166410141087185343849870143, 10.41466483395847907050266089360

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