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Properties

Label 2-483-7.2-c1-0-15
Degree $2$
Conductor $483$
Sign $0.790 + 0.612i$
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  + (0.838 − 1.45i)2-s + (−0.5 − 0.866i)3-s + (−0.407 − 0.705i)4-s + (−1.47 + 2.55i)5-s − 1.67·6-s + (2.59 + 0.523i)7-s + 1.98·8-s + (−0.499 + 0.866i)9-s + (2.47 + 4.28i)10-s + (1.83 + 3.17i)11-s + (−0.407 + 0.705i)12-s + 1.91·13-s + (2.93 − 3.32i)14-s + 2.94·15-s + (2.48 − 4.30i)16-s + (−1.51 − 2.62i)17-s + ⋯
L(s)  = 1  + (0.593 − 1.02i)2-s + (−0.288 − 0.499i)3-s + (−0.203 − 0.352i)4-s + (−0.659 + 1.14i)5-s − 0.684·6-s + (0.980 + 0.197i)7-s + 0.702·8-s + (−0.166 + 0.288i)9-s + (0.782 + 1.35i)10-s + (0.552 + 0.956i)11-s + (−0.117 + 0.203i)12-s + 0.531·13-s + (0.784 − 0.889i)14-s + 0.761·15-s + (0.620 − 1.07i)16-s + (−0.367 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80888 - 0.619044i\)
\(L(\frac12)\) \(\approx\) \(1.80888 - 0.619044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.59 - 0.523i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.838 + 1.45i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.47 - 2.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.83 - 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.91T + 13T^{2} \)
17 \( 1 + (1.51 + 2.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.08 - 1.87i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 0.852T + 29T^{2} \)
31 \( 1 + (1.01 + 1.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.71 + 4.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 + (5.86 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.40 - 5.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.54 + 9.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.08 + 1.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.74 + 9.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (5.11 + 8.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.20 + 3.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + (8.09 - 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.01T + 97T^{2} \)
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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

−10.99400980083154734337865413937, −10.68661645617296948431999577113, −9.294608993918482076741027525441, −7.82098822901113553617227641848, −7.37061625700045166285811774970, −6.27117419577665528419354377289, −4.80491470094578765155174587302, −3.95007278143970446374916922075, −2.72548155210506558229773144834, −1.66297058526405913660193688408, 1.22940701465319208422807259597, 3.87110023367980392610643318347, 4.51878272175353978422545041211, 5.35607997650425964367766051626, 6.20881280608406519515443898912, 7.37344247740563637114134399381, 8.490037392434956537341157498721, 8.721562754871115144297612569692, 10.34980827105858025309885529205, 11.22112360865625062526500071625

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