Properties

Label 2-483-161.90-c1-0-18
Degree $2$
Conductor $483$
Sign $-0.684 + 0.728i$
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  + (−1.19 − 0.769i)2-s + (0.989 − 0.142i)3-s + (0.0103 + 0.0227i)4-s + (−1.85 − 0.543i)5-s + (−1.29 − 0.591i)6-s + (2.00 + 1.72i)7-s + (−0.399 + 2.78i)8-s + (0.959 − 0.281i)9-s + (1.79 + 2.07i)10-s + (−1.95 − 3.04i)11-s + (0.0135 + 0.0210i)12-s + (2.98 − 2.58i)13-s + (−1.07 − 3.60i)14-s + (−1.90 − 0.274i)15-s + (2.65 − 3.06i)16-s + (2.49 − 5.45i)17-s + ⋯
L(s)  = 1  + (−0.846 − 0.544i)2-s + (0.571 − 0.0821i)3-s + (0.00519 + 0.0113i)4-s + (−0.827 − 0.242i)5-s + (−0.528 − 0.241i)6-s + (0.758 + 0.651i)7-s + (−0.141 + 0.983i)8-s + (0.319 − 0.0939i)9-s + (0.568 + 0.655i)10-s + (−0.590 − 0.918i)11-s + (0.00390 + 0.00607i)12-s + (0.827 − 0.717i)13-s + (−0.287 − 0.964i)14-s + (−0.492 − 0.0708i)15-s + (0.662 − 0.765i)16-s + (0.604 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.684 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315709 - 0.729959i\)
\(L(\frac12)\) \(\approx\) \(0.315709 - 0.729959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (-2.00 - 1.72i)T \)
23 \( 1 + (4.57 - 1.44i)T \)
good2 \( 1 + (1.19 + 0.769i)T + (0.830 + 1.81i)T^{2} \)
5 \( 1 + (1.85 + 0.543i)T + (4.20 + 2.70i)T^{2} \)
11 \( 1 + (1.95 + 3.04i)T + (-4.56 + 10.0i)T^{2} \)
13 \( 1 + (-2.98 + 2.58i)T + (1.85 - 12.8i)T^{2} \)
17 \( 1 + (-2.49 + 5.45i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (2.11 + 4.62i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (0.273 - 0.599i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (9.09 + 1.30i)T + (29.7 + 8.73i)T^{2} \)
37 \( 1 + (-1.49 - 5.10i)T + (-31.1 + 20.0i)T^{2} \)
41 \( 1 + (0.999 - 3.40i)T + (-34.4 - 22.1i)T^{2} \)
43 \( 1 + (-11.5 + 1.65i)T + (41.2 - 12.1i)T^{2} \)
47 \( 1 + 8.64iT - 47T^{2} \)
53 \( 1 + (4.59 + 3.98i)T + (7.54 + 52.4i)T^{2} \)
59 \( 1 + (-2.75 + 2.39i)T + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (-1.18 + 8.21i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (2.51 - 3.92i)T + (-27.8 - 60.9i)T^{2} \)
71 \( 1 + (-2.52 - 1.62i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-1.80 + 0.824i)T + (47.8 - 55.1i)T^{2} \)
79 \( 1 + (-8.73 + 7.56i)T + (11.2 - 78.1i)T^{2} \)
83 \( 1 + (-1.39 + 0.410i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (0.896 + 6.23i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-9.51 - 2.79i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.77689412930717880758377881320, −9.603185887310901434027930952223, −8.785740109099573799466845601456, −8.185662403277643159483227789876, −7.60603299337833489289886122200, −5.80821757495002240533354415634, −4.92634341730601749158194309503, −3.39776929641527341811805856068, −2.23847415182015011807538487328, −0.61953603164120456026797026710, 1.72191306532352271631635644543, 3.86769500990330959345066902038, 4.08846077531618791838968700486, 6.00868176128450113674282452074, 7.32413219797697031346052437946, 7.75780519571154132441606601306, 8.350304287001042325575677794495, 9.323904596694091729801111732581, 10.35389357494811315658867454084, 10.94727931631280125726699508600

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