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Properties

Label 2-483-161.4-c1-0-10
Degree $2$
Conductor $483$
Sign $0.109 - 0.994i$
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  + (−0.957 + 1.34i)2-s + (−0.235 + 0.971i)3-s + (−0.236 − 0.684i)4-s + (0.115 − 2.42i)5-s + (−1.08 − 1.24i)6-s + (2.21 − 1.43i)7-s + (−2.02 − 0.593i)8-s + (−0.888 − 0.458i)9-s + (3.15 + 2.47i)10-s + (−0.243 − 0.341i)11-s + (0.720 − 0.0688i)12-s + (−0.871 + 6.05i)13-s + (−0.189 + 4.36i)14-s + (2.33 + 0.684i)15-s + (3.87 − 3.04i)16-s + (5.69 − 1.09i)17-s + ⋯
L(s)  = 1  + (−0.676 + 0.950i)2-s + (−0.136 + 0.561i)3-s + (−0.118 − 0.342i)4-s + (0.0516 − 1.08i)5-s + (−0.441 − 0.509i)6-s + (0.838 − 0.544i)7-s + (−0.714 − 0.209i)8-s + (−0.296 − 0.152i)9-s + (0.996 + 0.783i)10-s + (−0.0734 − 0.103i)11-s + (0.208 − 0.0198i)12-s + (−0.241 + 1.68i)13-s + (−0.0505 + 1.16i)14-s + (0.601 + 0.176i)15-s + (0.967 − 0.760i)16-s + (1.38 − 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.109 - 0.994i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.109 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781401 + 0.700147i\)
\(L(\frac12)\) \(\approx\) \(0.781401 + 0.700147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (-2.21 + 1.43i)T \)
23 \( 1 + (-3.96 - 2.69i)T \)
good2 \( 1 + (0.957 - 1.34i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (-0.115 + 2.42i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (0.243 + 0.341i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (0.871 - 6.05i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-5.69 + 1.09i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (-4.77 - 0.921i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (0.408 + 0.471i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (5.22 - 4.98i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (-6.81 - 3.51i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (-3.15 - 2.02i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (0.780 - 0.229i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (4.23 + 7.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.64 + 1.45i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (2.86 + 2.25i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (2.90 + 11.9i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (-11.4 - 1.09i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (-5.27 - 11.5i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (1.53 + 4.42i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-0.449 + 0.179i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (7.46 - 4.79i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (2.05 + 1.96i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (5.73 + 3.68i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.27184604331916961326997286925, −9.779448086166188005710658833703, −9.352073465137809024552653018378, −8.461314108104272531335771813883, −7.67444823596726341146995081464, −6.82837052165995885677193917467, −5.45041843230342653961243985808, −4.82203192598509763276371127912, −3.52241455784931394760487013144, −1.21891776612918093923685589536, 1.05768444198315758994839658686, 2.52665922973584617605716023692, 3.17587055800583406042292143618, 5.33616677747538597923861189608, 6.00126973848646170883363623904, 7.48608438646589085991477968520, 7.979389492839746696241565156835, 9.201516145300641696362650749560, 10.12088109119227222338671196145, 10.87140736258822298273548196716

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