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Properties

Label 2-483-21.20-c1-0-39
Degree $2$
Conductor $483$
Sign $0.553 + 0.832i$
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  − i·2-s + (−0.923 + 1.46i)3-s + 4-s + 3.37·5-s + (1.46 + 0.923i)6-s + (−2.41 − 1.08i)7-s − 3i·8-s + (−1.29 − 2.70i)9-s − 3.37i·10-s − 3.41i·11-s + (−0.923 + 1.46i)12-s − 3.69i·13-s + (−1.08 + 2.41i)14-s + (−3.12 + 4.94i)15-s − 16-s + 4.46·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.533 + 0.845i)3-s + 0.5·4-s + 1.51·5-s + (0.598 + 0.377i)6-s + (−0.912 − 0.409i)7-s − 1.06i·8-s + (−0.430 − 0.902i)9-s − 1.06i·10-s − 1.02i·11-s + (−0.266 + 0.422i)12-s − 1.02i·13-s + (−0.289 + 0.645i)14-s + (−0.805 + 1.27i)15-s − 0.250·16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44019 - 0.771968i\)
\(L(\frac12)\) \(\approx\) \(1.44019 - 0.771968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 - 1.46i)T \)
7 \( 1 + (2.41 + 1.08i)T \)
23 \( 1 + iT \)
good2 \( 1 + iT - 2T^{2} \)
5 \( 1 - 3.37T + 5T^{2} \)
11 \( 1 + 3.41iT - 11T^{2} \)
13 \( 1 + 3.69iT - 13T^{2} \)
17 \( 1 - 4.46T + 17T^{2} \)
19 \( 1 - 5.22iT - 19T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 4.90iT - 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + 8.28T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 + 5.99T + 47T^{2} \)
53 \( 1 + 0.828iT - 53T^{2} \)
59 \( 1 - 8.60T + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 1.65iT - 71T^{2} \)
73 \( 1 - 6.75iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 0.634T + 83T^{2} \)
89 \( 1 + 8.79T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−10.56643290773511368641985558320, −10.00381917006924993058031325515, −9.791169463507412224680587064037, −8.417223232170090654604758280018, −6.77519236287647788160030189130, −5.98720759945273087405755287062, −5.40626646246050736530396109390, −3.56338865766601545154175217222, −2.96766282924794599627515445771, −1.14233022559293502845743716813, 1.82110370656645442699911043351, 2.60991650211926481388931996972, 5.01486385096143115054246076563, 5.82851723895709319452565312397, 6.57471725878394483275005167458, 6.99769173352471527306541924980, 8.189869589588143388085048518287, 9.507488414621832086759287530045, 9.980757728151184828732082970712, 11.32278246655066630360338164521

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