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Properties

Label 2-483-161.19-c1-0-29
Degree $2$
Conductor $483$
Sign $0.787 + 0.616i$
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  + (1.63 + 0.155i)2-s + (0.690 − 0.723i)3-s + (0.670 + 0.129i)4-s + (3.55 − 1.83i)5-s + (1.23 − 1.07i)6-s + (−2.60 + 0.462i)7-s + (−2.07 − 0.607i)8-s + (−0.0475 − 0.998i)9-s + (6.07 − 2.43i)10-s + (0.0284 + 0.298i)11-s + (0.555 − 0.395i)12-s + (5.98 + 0.861i)13-s + (−4.31 + 0.347i)14-s + (1.12 − 3.83i)15-s + (−4.54 − 1.82i)16-s + (−0.680 + 1.96i)17-s + ⋯
L(s)  = 1  + (1.15 + 0.110i)2-s + (0.398 − 0.417i)3-s + (0.335 + 0.0645i)4-s + (1.58 − 0.819i)5-s + (0.505 − 0.437i)6-s + (−0.984 + 0.174i)7-s + (−0.731 − 0.214i)8-s + (−0.0158 − 0.332i)9-s + (1.92 − 0.769i)10-s + (0.00858 + 0.0899i)11-s + (0.160 − 0.114i)12-s + (1.66 + 0.238i)13-s + (−1.15 + 0.0929i)14-s + (0.290 − 0.990i)15-s + (−1.13 − 0.455i)16-s + (−0.165 + 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85771 - 0.985749i\)
\(L(\frac12)\) \(\approx\) \(2.85771 - 0.985749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.690 + 0.723i)T \)
7 \( 1 + (2.60 - 0.462i)T \)
23 \( 1 + (3.35 - 3.42i)T \)
good2 \( 1 + (-1.63 - 0.155i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-3.55 + 1.83i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (-0.0284 - 0.298i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-5.98 - 0.861i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.680 - 1.96i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (0.499 + 1.44i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-1.92 - 2.22i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (6.99 - 1.69i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (-1.65 + 0.0786i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (3.47 - 5.40i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-3.04 - 10.3i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (-7.29 - 4.21i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.14 + 5.27i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (5.21 + 13.0i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (1.47 - 1.41i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (-2.46 - 1.75i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (2.43 + 5.34i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (0.874 - 4.53i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (4.61 - 5.87i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (-1.72 + 1.11i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (0.749 - 3.08i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (9.65 + 6.20i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.00651823000777120255194056521, −9.656103588965042023268773719704, −9.236586468743313097899308401590, −8.396838654881403863822993382641, −6.58482978363983852949901599691, −6.12502567467740899437036662006, −5.39188830758469679634505713332, −4.08781537940787265573725617026, −2.97740980536600146756419846576, −1.56558405077017634934205197652, 2.28715302609709160238361559602, 3.24824092327033979362805262872, 4.07291456066836942519891708067, 5.69354586983749668846139303215, 5.98031334201271974287198439744, 6.97517870792921114252125920681, 8.731444150656378103879612921600, 9.344995724338269129094565298489, 10.34948631838569413968130384517, 10.85611689354892326614693529261

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