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Properties

Label 2-483-21.20-c1-0-25
Degree $2$
Conductor $483$
Sign $0.975 - 0.218i$
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  + 0.618i·2-s + (−1.61 + 0.618i)3-s + 1.61·4-s − 0.381·5-s + (−0.381 − 1.00i)6-s + (0.381 − 2.61i)7-s + 2.23i·8-s + (2.23 − 2.00i)9-s − 0.236i·10-s − 2.23i·11-s + (−2.61 + 1.00i)12-s − 0.145i·13-s + (1.61 + 0.236i)14-s + (0.618 − 0.236i)15-s + 1.85·16-s + 7.47·17-s + ⋯
L(s)  = 1  + 0.437i·2-s + (−0.934 + 0.356i)3-s + 0.809·4-s − 0.170·5-s + (−0.155 − 0.408i)6-s + (0.144 − 0.989i)7-s + 0.790i·8-s + (0.745 − 0.666i)9-s − 0.0746i·10-s − 0.674i·11-s + (−0.755 + 0.288i)12-s − 0.0404i·13-s + (0.432 + 0.0630i)14-s + (0.159 − 0.0609i)15-s + 0.463·16-s + 1.81·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.975 - 0.218i$
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: \((2,\ 483,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32199 + 0.146001i\)
\(L(\frac12)\) \(\approx\) \(1.32199 + 0.146001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 - 0.618i)T \)
7 \( 1 + (-0.381 + 2.61i)T \)
23 \( 1 - iT \)
good2 \( 1 - 0.618iT - 2T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + 0.145iT - 13T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 + 3.47iT - 19T^{2} \)
29 \( 1 + 6.23iT - 29T^{2} \)
31 \( 1 - 9.47iT - 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 4.23T + 41T^{2} \)
43 \( 1 - 5.38T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 6.56iT - 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 + 5.85iT - 61T^{2} \)
67 \( 1 + 8.38T + 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 3.76iT - 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−10.95357402568123742691886062457, −10.42099979398851560350363506179, −9.415460889169281124981059391306, −7.85384029749001926808225191068, −7.39312133864889674666895858711, −6.24950225500305866874104666780, −5.61698431371832350317012482678, −4.40040607583075241343838917013, −3.18407411443623148358256842240, −1.07632849649512588977219094797, 1.41308452259816050191723358517, 2.60044587185931908194168802920, 4.13164799423517797233267763376, 5.64323863464469169906565641734, 6.03698500824422165967690122463, 7.38960518737465808985714979799, 7.88036549664719796623850685206, 9.549021218861575096191413617821, 10.19770632903583506617896661448, 11.19077743755203574287951455249

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