Properties

Label 2-483-23.8-c1-0-1
Degree $2$
Conductor $483$
Sign $-0.764 + 0.644i$
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  + (−0.317 + 2.20i)2-s + (0.415 + 0.909i)3-s + (−2.84 − 0.835i)4-s + (0.0577 + 0.0666i)5-s + (−2.13 + 0.627i)6-s + (0.841 + 0.540i)7-s + (0.895 − 1.96i)8-s + (−0.654 + 0.755i)9-s + (−0.165 + 0.106i)10-s + (0.345 + 2.40i)11-s + (−0.422 − 2.93i)12-s + (−4.37 + 2.80i)13-s + (−1.45 + 1.68i)14-s + (−0.0366 + 0.0801i)15-s + (−0.950 − 0.611i)16-s + (0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (−0.224 + 1.55i)2-s + (0.239 + 0.525i)3-s + (−1.42 − 0.417i)4-s + (0.0258 + 0.0297i)5-s + (−0.873 + 0.256i)6-s + (0.317 + 0.204i)7-s + (0.316 − 0.693i)8-s + (−0.218 + 0.251i)9-s + (−0.0522 + 0.0335i)10-s + (0.104 + 0.723i)11-s + (−0.121 − 0.847i)12-s + (−1.21 + 0.779i)13-s + (−0.390 + 0.450i)14-s + (−0.00945 + 0.0207i)15-s + (−0.237 − 0.152i)16-s + (0.232 − 0.0683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351588 - 0.962742i\)
\(L(\frac12)\) \(\approx\) \(0.351588 - 0.962742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (2.79 - 3.89i)T \)
good2 \( 1 + (0.317 - 2.20i)T + (-1.91 - 0.563i)T^{2} \)
5 \( 1 + (-0.0577 - 0.0666i)T + (-0.711 + 4.94i)T^{2} \)
11 \( 1 + (-0.345 - 2.40i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (4.37 - 2.80i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-0.959 + 0.281i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (2.58 + 0.759i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-6.21 + 1.82i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.830 - 1.81i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (2.36 - 2.72i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (-8.10 - 9.35i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (0.893 + 1.95i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 0.184T + 47T^{2} \)
53 \( 1 + (-4.77 - 3.06i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-6.09 + 3.91i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (3.16 - 6.92i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-0.0438 + 0.305i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-0.152 + 1.06i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.04 - 0.600i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-9.37 + 6.02i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (4.90 - 5.65i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (6.22 + 13.6i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (6.53 + 7.53i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.57212896726027489320590578769, −10.17450696255618691900658444789, −9.536991954776980741302217897487, −8.664428681909559504128254488872, −7.84079033882538675330640899379, −7.02303364846420006721595024730, −6.11243297009020974493632927109, −4.93835541866974959962854212392, −4.36344916912005152587550068752, −2.41152361564146760075562604235, 0.64207815718958589876177653686, 2.09792723537614995362209478538, 3.05869598070317502259375693018, 4.19869129728264131527284651484, 5.51771090075575303043407508612, 6.90309052256664233751677537337, 8.030697080812407247858734294850, 8.800194145882819228832574796793, 9.790444547234663911363984594951, 10.58123635819080324220113722516

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