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Properties

Label 2-483-7.2-c1-0-24
Degree $2$
Conductor $483$
Sign $0.853 + 0.520i$
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  + (0.384 − 0.666i)2-s + (0.5 + 0.866i)3-s + (0.704 + 1.21i)4-s + (2.01 − 3.48i)5-s + 0.769·6-s + (2.63 + 0.229i)7-s + 2.62·8-s + (−0.499 + 0.866i)9-s + (−1.54 − 2.68i)10-s + (−2.73 − 4.73i)11-s + (−0.704 + 1.21i)12-s − 4.39·13-s + (1.16 − 1.66i)14-s + 4.02·15-s + (−0.400 + 0.693i)16-s + (1.12 + 1.94i)17-s + ⋯
L(s)  = 1  + (0.271 − 0.470i)2-s + (0.288 + 0.499i)3-s + (0.352 + 0.609i)4-s + (0.900 − 1.55i)5-s + 0.313·6-s + (0.996 + 0.0869i)7-s + 0.926·8-s + (−0.166 + 0.288i)9-s + (−0.489 − 0.848i)10-s + (−0.824 − 1.42i)11-s + (−0.203 + 0.352i)12-s − 1.22·13-s + (0.311 − 0.445i)14-s + 1.03·15-s + (−0.100 + 0.173i)16-s + (0.272 + 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.853 + 0.520i$
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: \((2,\ 483,\ (\ :1/2),\ 0.853 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23457 - 0.627146i\)
\(L(\frac12)\) \(\approx\) \(2.23457 - 0.627146i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.229i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.384 + 0.666i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.01 + 3.48i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.39T + 13T^{2} \)
17 \( 1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.78 - 3.08i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 + (-5.27 - 9.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.96 + 3.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.97T + 41T^{2} \)
43 \( 1 - 2.61T + 43T^{2} \)
47 \( 1 + (-0.259 + 0.448i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.31 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.421 + 0.729i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.50 - 7.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.699 - 1.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.10T + 71T^{2} \)
73 \( 1 + (2.66 + 4.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.39T + 83T^{2} \)
89 \( 1 + (6.50 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.35T + 97T^{2} \)
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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

−10.81937229482374700573796533856, −10.19832490282836883637677437257, −8.957394700229058229912559001018, −8.354656330801731421476182339175, −7.66505023399113575891588631205, −5.77876587610216059393166598248, −5.06800388894604336749079681111, −4.18741426107551312443360355655, −2.73774796274350885087814861637, −1.57930549792747379789326483748, 2.09150977352406282340023226080, 2.46635395335930269644316932787, 4.64843212036842376529270020249, 5.52751921751530350547925068332, 6.64002463870028373042327833011, 7.27628012025489260911880445384, 7.79339289309590029943594601676, 9.688826027328971162399741948440, 10.06694496879369892931255611068, 10.98375881139965958837906642997

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