Properties

Label 2-483-161.61-c1-0-9
Degree $2$
Conductor $483$
Sign $-0.133 - 0.991i$
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  + (1.28 + 1.01i)2-s + (−0.814 − 0.580i)3-s + (0.158 + 0.654i)4-s + (−2.14 + 0.413i)5-s + (−0.460 − 1.56i)6-s + (0.00533 + 2.64i)7-s + (0.901 − 1.97i)8-s + (0.327 + 0.945i)9-s + (−3.17 − 1.63i)10-s + (3.99 + 5.08i)11-s + (0.250 − 0.625i)12-s + (1.48 + 2.31i)13-s + (−2.66 + 3.40i)14-s + (1.98 + 0.908i)15-s + (4.34 − 2.24i)16-s + (1.48 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.908 + 0.714i)2-s + (−0.470 − 0.334i)3-s + (0.0793 + 0.327i)4-s + (−0.960 + 0.185i)5-s + (−0.188 − 0.640i)6-s + (0.00201 + 0.999i)7-s + (0.318 − 0.697i)8-s + (0.109 + 0.315i)9-s + (−1.00 − 0.518i)10-s + (1.20 + 1.53i)11-s + (0.0722 − 0.180i)12-s + (0.411 + 0.640i)13-s + (−0.712 + 0.910i)14-s + (0.513 + 0.234i)15-s + (1.08 − 0.560i)16-s + (0.359 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.133 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06846 + 1.22193i\)
\(L(\frac12)\) \(\approx\) \(1.06846 + 1.22193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (-0.00533 - 2.64i)T \)
23 \( 1 + (4.77 - 0.392i)T \)
good2 \( 1 + (-1.28 - 1.01i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (2.14 - 0.413i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-3.99 - 5.08i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-1.48 - 2.31i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.48 - 1.41i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (5.20 - 4.96i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-7.63 + 2.24i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.702 + 7.35i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (2.95 - 1.02i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-1.77 + 1.53i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (-4.43 + 2.02i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-8.32 - 4.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.16 - 0.246i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (0.385 - 0.746i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (3.00 + 4.21i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (2.20 + 5.49i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (0.209 - 1.45i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (3.38 - 0.821i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (2.66 - 0.126i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (1.91 - 2.20i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-3.72 - 0.355i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (2.30 + 2.65i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.72256975009889526215263501229, −10.39097383958917459759407915812, −9.442765860376725156442430694543, −8.194949689555355517246911077412, −7.30325823417727619004715669394, −6.37055486054829008346582263266, −5.85383982597748211269413033431, −4.37229889843798970252494500055, −4.00669943022713002151989452883, −1.86437598733057057409680688779, 0.850111357981188715319102072404, 3.15141552474582344758592179813, 3.93256008997156344220744047070, 4.53837042047182968219813451850, 5.81296030585852657286780680451, 6.89958585600484410203978605852, 8.169271744795522926200846314708, 8.821083047114888640780875381605, 10.51736055661984425066263199286, 10.84502972870548601801521418988

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