Properties

Label 2-483-161.61-c1-0-30
Degree $2$
Conductor $483$
Sign $-0.0447 + 0.999i$
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  + (1.54 + 1.21i)2-s + (−0.814 − 0.580i)3-s + (0.436 + 1.80i)4-s + (−4.04 + 0.779i)5-s + (−0.552 − 1.88i)6-s + (0.0404 − 2.64i)7-s + (0.120 − 0.263i)8-s + (0.327 + 0.945i)9-s + (−7.18 − 3.70i)10-s + (−3.14 − 3.99i)11-s + (0.688 − 1.71i)12-s + (0.309 + 0.481i)13-s + (3.27 − 4.03i)14-s + (3.74 + 1.71i)15-s + (3.79 − 1.95i)16-s + (−2.05 − 1.95i)17-s + ⋯
L(s)  = 1  + (1.09 + 0.857i)2-s + (−0.470 − 0.334i)3-s + (0.218 + 0.900i)4-s + (−1.80 + 0.348i)5-s + (−0.225 − 0.768i)6-s + (0.0152 − 0.999i)7-s + (0.0425 − 0.0931i)8-s + (0.109 + 0.315i)9-s + (−2.27 − 1.17i)10-s + (−0.948 − 1.20i)11-s + (0.198 − 0.496i)12-s + (0.0857 + 0.133i)13-s + (0.874 − 1.07i)14-s + (0.967 + 0.441i)15-s + (0.949 − 0.489i)16-s + (−0.497 − 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0447 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0447 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.0447 + 0.999i$
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: \((2,\ 483,\ (\ :1/2),\ -0.0447 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528560 - 0.552743i\)
\(L(\frac12)\) \(\approx\) \(0.528560 - 0.552743i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0404 + 2.64i)T \)
23 \( 1 + (4.78 + 0.380i)T \)
good2 \( 1 + (-1.54 - 1.21i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (4.04 - 0.779i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (3.14 + 3.99i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-0.309 - 0.481i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (2.05 + 1.95i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (3.00 - 2.86i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (1.31 - 0.386i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.329 - 3.44i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-7.06 + 2.44i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-5.88 + 5.09i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (10.9 - 4.98i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (10.5 + 6.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.07 + 0.0511i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (0.353 - 0.685i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (2.47 + 3.48i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-3.80 - 9.49i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (0.213 - 1.48i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-8.43 + 2.04i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (0.678 - 0.0323i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-5.76 + 6.65i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-16.5 - 1.58i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (9.64 + 11.1i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.03491508537525026577255043879, −10.25600329011914439635608455993, −8.236393157405680628437790931206, −7.80796633342354395522063501810, −6.95291356109054785514450505468, −6.19895371157103009291791890150, −4.94728006518665483268898911700, −4.08810194302484669845814391797, −3.31265359114994871329202190592, −0.33532330234799028958652958686, 2.27981849203600557054325996579, 3.53056262746202864865413558257, 4.57091953415840946358536411764, 4.90025217818399818609389833082, 6.26660113211752459430212560459, 7.76314776129614763070662522548, 8.351109849214660505787527967616, 9.704081480281930888898409765529, 10.87149500314238296990117208360, 11.41337978074179032190240162290

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