Properties

Label 2-483-161.18-c1-0-6
Degree $2$
Conductor $483$
Sign $-0.529 + 0.848i$
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  + (0.523 + 2.15i)2-s + (0.327 + 0.945i)3-s + (−2.60 + 1.34i)4-s + (−2.17 + 0.870i)5-s + (−1.86 + 1.20i)6-s + (−1.34 + 2.27i)7-s + (−1.35 − 1.56i)8-s + (−0.786 + 0.618i)9-s + (−3.01 − 4.23i)10-s + (0.945 − 3.89i)11-s + (−2.12 − 2.02i)12-s + (0.275 − 0.603i)13-s + (−5.62 − 1.70i)14-s + (−1.53 − 1.76i)15-s + (−0.739 + 1.03i)16-s + (0.351 + 7.38i)17-s + ⋯
L(s)  = 1  + (0.370 + 1.52i)2-s + (0.188 + 0.545i)3-s + (−1.30 + 0.671i)4-s + (−0.972 + 0.389i)5-s + (−0.762 + 0.490i)6-s + (−0.508 + 0.861i)7-s + (−0.478 − 0.551i)8-s + (−0.262 + 0.206i)9-s + (−0.953 − 1.33i)10-s + (0.285 − 1.17i)11-s + (−0.612 − 0.583i)12-s + (0.0764 − 0.167i)13-s + (−1.50 − 0.456i)14-s + (−0.395 − 0.457i)15-s + (−0.184 + 0.259i)16-s + (0.0853 + 1.79i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534147 - 0.963283i\)
\(L(\frac12)\) \(\approx\) \(0.534147 - 0.963283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (1.34 - 2.27i)T \)
23 \( 1 + (-4.79 + 0.0718i)T \)
good2 \( 1 + (-0.523 - 2.15i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (2.17 - 0.870i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-0.945 + 3.89i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (-0.275 + 0.603i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.351 - 7.38i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.332 + 6.98i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-0.114 + 0.0733i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (8.14 - 1.56i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (-1.24 + 0.980i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (1.77 - 12.3i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (3.56 - 4.10i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (-4.50 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.01 - 0.574i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (-1.62 - 2.28i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (-2.77 + 8.02i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (5.86 - 5.59i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-11.0 - 3.24i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (4.32 - 2.23i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (-16.0 + 1.53i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (-1.07 - 7.46i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (12.4 + 2.40i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (1.33 - 9.27i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.34260622014539234253881394242, −10.90772533638593468233515396057, −9.312965168180676928827270598999, −8.611659501914689386091923850532, −7.976193492321352920525155018045, −6.85175620081486798849198486777, −6.07433879979445500120570579230, −5.18282100181614142600059321854, −3.98314190915832520539662496585, −3.08872125171402988234805624233, 0.58752828632273175775537481938, 2.00190990595906991689978945657, 3.45497241672045321845380814161, 4.07437941149919126811046733634, 5.17756352207271955922716936198, 7.09325850006881304188897650943, 7.44251744862915370764323028816, 8.916587230065581285357490051057, 9.741076770347223950286112857010, 10.53684099173461551689340900178

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