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Properties

Label 2-483-7.4-c1-0-7
Degree $2$
Conductor $483$
Sign $-0.266 - 0.963i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + (1.5 + 2.59i)5-s − 0.999·6-s + (−0.5 + 2.59i)7-s + 3·8-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)10-s + (1 − 1.73i)11-s + (0.499 + 0.866i)12-s − 13-s + (−2.5 + 0.866i)14-s − 3·15-s + (0.500 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.670 + 1.16i)5-s − 0.408·6-s + (−0.188 + 0.981i)7-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.474 + 0.821i)10-s + (0.301 − 0.522i)11-s + (0.144 + 0.249i)12-s − 0.277·13-s + (−0.668 + 0.231i)14-s − 0.774·15-s + (0.125 + 0.216i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13234 + 1.48844i\)
\(L(\frac12)\) \(\approx\) \(1.13234 + 1.48844i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−11.03633116559666662156276817469, −10.42096863145797375827484525849, −9.639121804605024141459786475728, −8.624407323596410742243285903787, −7.18928450642202972307256093786, −6.37439891964091385971059755362, −5.81846973557231875627856916160, −4.92445290457440851689817339709, −3.33962610718409731458706555556, −2.09434948118240671861024327379, 1.15904042009298692192322873976, 2.32543613246226100827491627562, 3.95428314064053294690307582548, 4.76345321644918578195391499919, 5.96780917188122952596228672618, 7.19180650265218594247773685345, 7.79180852060727552906574294586, 9.083930644149681039722741371596, 9.928104513981145298815537667749, 10.92035146742730266923296917944

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