Properties

Label 2-483-161.19-c1-0-14
Degree $2$
Conductor $483$
Sign $0.107 + 0.994i$
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  + (−0.922 − 0.0880i)2-s + (0.690 − 0.723i)3-s + (−1.12 − 0.216i)4-s + (−2.93 + 1.51i)5-s + (−0.700 + 0.606i)6-s + (0.614 + 2.57i)7-s + (2.79 + 0.819i)8-s + (−0.0475 − 0.998i)9-s + (2.83 − 1.13i)10-s + (−0.394 − 4.13i)11-s + (−0.930 + 0.662i)12-s + (1.09 + 0.157i)13-s + (−0.340 − 2.42i)14-s + (−0.929 + 3.16i)15-s + (−0.382 − 0.153i)16-s + (1.79 − 5.19i)17-s + ⋯
L(s)  = 1  + (−0.652 − 0.0622i)2-s + (0.398 − 0.417i)3-s + (−0.560 − 0.108i)4-s + (−1.31 + 0.675i)5-s + (−0.285 + 0.247i)6-s + (0.232 + 0.972i)7-s + (0.987 + 0.289i)8-s + (−0.0158 − 0.332i)9-s + (0.896 − 0.359i)10-s + (−0.118 − 1.24i)11-s + (−0.268 + 0.191i)12-s + (0.303 + 0.0435i)13-s + (−0.0909 − 0.648i)14-s + (−0.239 + 0.817i)15-s + (−0.0955 − 0.0382i)16-s + (0.435 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.107 + 0.994i$
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: \((2,\ 483,\ (\ :1/2),\ 0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467908 - 0.420021i\)
\(L(\frac12)\) \(\approx\) \(0.467908 - 0.420021i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.614 - 2.57i)T \)
23 \( 1 + (-0.291 + 4.78i)T \)
good2 \( 1 + (0.922 + 0.0880i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (2.93 - 1.51i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.394 + 4.13i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-1.09 - 0.157i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.79 + 5.19i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (0.394 + 1.13i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (3.35 + 3.87i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-1.56 + 0.379i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (-5.81 + 0.277i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (3.60 - 5.61i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (2.08 + 7.10i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (-11.3 - 6.52i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.30 + 2.92i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (4.42 + 11.0i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-8.09 + 7.72i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (-8.37 - 5.96i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (2.27 + 4.97i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.174 + 0.906i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-1.81 + 2.30i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (-12.0 + 7.75i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (0.491 - 2.02i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (4.31 + 2.77i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.97211871263899661745120149242, −9.666278053263911783978803236746, −8.780477463493722299690060674233, −8.157688325713106487462022347089, −7.54572047693171766714802217566, −6.29734671538895978940681563919, −5.00625385908704067151609278136, −3.70817932242498907944196161280, −2.61628800197738385174840529795, −0.53539403424711378315599361922, 1.31245982182947970844472577994, 3.84050692871931279504141825386, 4.11630293450309021362393650112, 5.17261371919191391668399883481, 7.19729761998961188339239044533, 7.80510168980356648650303123442, 8.394719253010384201206109550577, 9.331476132251948239524191039725, 10.19435385156266919371832732530, 10.88550731273265606183141564628

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