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Properties

Label 2-483-7.2-c1-0-22
Degree $2$
Conductor $483$
Sign $0.192 + 0.981i$
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  + (−1.37 + 2.37i)2-s + (−0.5 − 0.866i)3-s + (−2.76 − 4.79i)4-s + (0.957 − 1.65i)5-s + 2.74·6-s + (1.61 + 2.09i)7-s + 9.71·8-s + (−0.499 + 0.866i)9-s + (2.62 + 4.55i)10-s + (−2.66 − 4.61i)11-s + (−2.76 + 4.79i)12-s − 4.78·13-s + (−7.20 + 0.946i)14-s − 1.91·15-s + (−7.80 + 13.5i)16-s + (−1.30 − 2.26i)17-s + ⋯
L(s)  = 1  + (−0.970 + 1.68i)2-s + (−0.288 − 0.499i)3-s + (−1.38 − 2.39i)4-s + (0.428 − 0.741i)5-s + 1.12·6-s + (0.608 + 0.793i)7-s + 3.43·8-s + (−0.166 + 0.288i)9-s + (0.831 + 1.43i)10-s + (−0.802 − 1.39i)11-s + (−0.799 + 1.38i)12-s − 1.32·13-s + (−1.92 + 0.252i)14-s − 0.494·15-s + (−1.95 + 3.37i)16-s + (−0.317 − 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.192 + 0.981i$
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: \((2,\ 483,\ (\ :1/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222061 - 0.182679i\)
\(L(\frac12)\) \(\approx\) \(0.222061 - 0.182679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 2.09i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.37 - 2.37i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.957 + 1.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.66 + 4.61i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.78T + 13T^{2} \)
17 \( 1 + (1.30 + 2.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.23 - 3.86i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 8.86T + 29T^{2} \)
31 \( 1 + (2.95 + 5.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.45 + 2.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.98T + 41T^{2} \)
43 \( 1 + 6.45T + 43T^{2} \)
47 \( 1 + (-3.47 + 6.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.506 + 0.876i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.07 + 5.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.12 - 1.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.69 + 8.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + (1.92 + 3.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.48 - 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.29T + 83T^{2} \)
89 \( 1 + (-5.04 + 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−10.50762647804009162364503723410, −9.380968967492432065890927148617, −8.827591380685649931113781684192, −7.959150059447623012424739486766, −7.38615530712445794259614899588, −6.01670323296644325328841142885, −5.54348393072248877544287307587, −4.83870067314414145652954231501, −1.93676231357575319697791355266, −0.23931177447362066706748913180, 1.88513425377977418165633511539, 2.80045330911632367989765544358, 4.22302773794366937142348451798, 4.91325795660983451818879159537, 7.11767903874474100982684869437, 7.66144230899863383815965760096, 8.966507840862387738076479994295, 9.819918749335363252220242585315, 10.39762699286813889872129243160, 10.83065327859134253145434378375

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