Properties

Label 2-483-161.2-c1-0-18
Degree $2$
Conductor $483$
Sign $0.949 + 0.313i$
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  + (0.189 − 0.0977i)2-s + (−0.786 + 0.618i)3-s + (−1.13 + 1.59i)4-s + (2.30 − 2.20i)5-s + (−0.0886 + 0.194i)6-s + (2.00 − 1.72i)7-s + (−0.120 + 0.834i)8-s + (0.235 − 0.971i)9-s + (0.222 − 0.643i)10-s + (−1.11 − 0.572i)11-s + (−0.0929 − 1.95i)12-s + (−1.60 − 1.85i)13-s + (0.212 − 0.522i)14-s + (−0.454 + 3.15i)15-s + (−1.21 − 3.52i)16-s + (5.71 − 0.545i)17-s + ⋯
L(s)  = 1  + (0.134 − 0.0691i)2-s + (−0.453 + 0.356i)3-s + (−0.566 + 0.796i)4-s + (1.03 − 0.984i)5-s + (−0.0361 + 0.0792i)6-s + (0.759 − 0.650i)7-s + (−0.0424 + 0.295i)8-s + (0.0785 − 0.323i)9-s + (0.0703 − 0.203i)10-s + (−0.334 − 0.172i)11-s + (−0.0268 − 0.563i)12-s + (−0.445 − 0.514i)13-s + (0.0567 − 0.139i)14-s + (−0.117 + 0.815i)15-s + (−0.304 − 0.880i)16-s + (1.38 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.949 + 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44133 - 0.232139i\)
\(L(\frac12)\) \(\approx\) \(1.44133 - 0.232139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (-2.00 + 1.72i)T \)
23 \( 1 + (-3.65 - 3.10i)T \)
good2 \( 1 + (-0.189 + 0.0977i)T + (1.16 - 1.62i)T^{2} \)
5 \( 1 + (-2.30 + 2.20i)T + (0.237 - 4.99i)T^{2} \)
11 \( 1 + (1.11 + 0.572i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (1.60 + 1.85i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-5.71 + 0.545i)T + (16.6 - 3.21i)T^{2} \)
19 \( 1 + (0.160 + 0.0153i)T + (18.6 + 3.59i)T^{2} \)
29 \( 1 + (-0.470 + 1.03i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-8.66 + 3.46i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (-0.170 + 0.700i)T + (-32.8 - 16.9i)T^{2} \)
41 \( 1 + (-4.70 - 1.38i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-1.06 - 7.37i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (2.15 - 3.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.54 + 1.26i)T + (49.2 + 19.6i)T^{2} \)
59 \( 1 + (4.38 - 12.6i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (8.92 + 7.01i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (-0.440 + 9.24i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (5.45 + 3.50i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.567 - 0.796i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (-3.35 + 0.645i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (12.4 - 3.66i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-2.61 - 1.04i)T + (64.4 + 61.4i)T^{2} \)
97 \( 1 + (8.44 + 2.47i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.93608901203010628484065883003, −9.873321491505527539246853426085, −9.347456060223914721026995286383, −8.170854060475761859843607122058, −7.58662026411927563760226214620, −5.92267204145032640708392802464, −5.04308777700785860179077986166, −4.48295054239527065957379316549, −3.01717283281107967674741787731, −1.09021407492254026357016597361, 1.51038036778930615933722015888, 2.71911772725688959965102693009, 4.68369057941625378179202900487, 5.48713374978424412052936660288, 6.20549890369813630016847972162, 7.11407014678643637825167883849, 8.374276438592879567293129208351, 9.467157375843350771421385229632, 10.24407748380383419618939413243, 10.80327159173649149896960366740

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