Properties

Label 2-483-161.34-c1-0-18
Degree $2$
Conductor $483$
Sign $0.691 + 0.722i$
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  + (−1.19 + 0.769i)2-s + (−0.989 − 0.142i)3-s + (0.0103 − 0.0227i)4-s + (1.85 − 0.543i)5-s + (1.29 − 0.591i)6-s + (1.99 − 1.74i)7-s + (−0.399 − 2.78i)8-s + (0.959 + 0.281i)9-s + (−1.79 + 2.07i)10-s + (−1.95 + 3.04i)11-s + (−0.0135 + 0.0210i)12-s + (−2.98 − 2.58i)13-s + (−1.04 + 3.61i)14-s + (−1.90 + 0.274i)15-s + (2.65 + 3.06i)16-s + (−2.49 − 5.45i)17-s + ⋯
L(s)  = 1  + (−0.846 + 0.544i)2-s + (−0.571 − 0.0821i)3-s + (0.00519 − 0.0113i)4-s + (0.827 − 0.242i)5-s + (0.528 − 0.241i)6-s + (0.752 − 0.658i)7-s + (−0.141 − 0.983i)8-s + (0.319 + 0.0939i)9-s + (−0.568 + 0.655i)10-s + (−0.590 + 0.918i)11-s + (−0.00390 + 0.00607i)12-s + (−0.827 − 0.717i)13-s + (−0.278 + 0.966i)14-s + (−0.492 + 0.0708i)15-s + (0.662 + 0.765i)16-s + (−0.604 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627591 - 0.267976i\)
\(L(\frac12)\) \(\approx\) \(0.627591 - 0.267976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (-1.99 + 1.74i)T \)
23 \( 1 + (4.57 + 1.44i)T \)
good2 \( 1 + (1.19 - 0.769i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (-1.85 + 0.543i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (1.95 - 3.04i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.98 + 2.58i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.49 + 5.45i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-2.11 + 4.62i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.273 + 0.599i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-9.09 + 1.30i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (-1.49 + 5.10i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-0.999 - 3.40i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-11.5 - 1.65i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 8.64iT - 47T^{2} \)
53 \( 1 + (4.59 - 3.98i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (2.75 + 2.39i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (1.18 + 8.21i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (2.51 + 3.92i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (-2.52 + 1.62i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (1.80 + 0.824i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-8.73 - 7.56i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (1.39 + 0.410i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.896 + 6.23i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (9.51 - 2.79i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.56525068845842240011851461498, −9.814874537993523146944510729455, −9.256757856760008715584879431410, −7.85816882726793065619476446836, −7.45528804773479990301024342097, −6.48511753888550226911470679355, −5.17997858917008233758112567310, −4.46664085172018075092040574964, −2.40995241930767455515153740351, −0.60451553732400795309915391459, 1.54963352800275006838340462511, 2.52089772336836827638956057170, 4.50820823828511540543555900700, 5.72613133009925317107145742900, 6.08389644891385890295104711670, 7.83184044837492615571942096348, 8.546775751548501158098739026624, 9.555930228496214535491639602701, 10.23125975598932996595371129238, 10.89562890610336561356581849366

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