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Properties

Label 2-483-21.20-c1-0-44
Degree $2$
Conductor $483$
Sign $0.872 + 0.487i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.872 + 0.487i$
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: \((2,\ 483,\ (\ :1/2),\ 0.872 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27330 - 0.592277i\)
\(L(\frac12)\) \(\approx\) \(2.27330 - 0.592277i\)
\(L(\frac{3}{2})\) 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

−10.80188246600214473316104821375, −10.13346020206698756148084100232, −9.298762806981352456913337530007, −8.248363481438303950267186450646, −7.14608232646074804621520514852, −6.70668093817699346898275281593, −4.82728574092827190671717886850, −3.92329811350257781320650975924, −2.73608755653770352116939033284, −1.67831645586395942662184152274, 1.98694529128153579213550212957, 2.68950323975525126104993337524, 4.17944931062128854471464480788, 5.88171342667040902915940334340, 6.32140639643068285653828306227, 7.58738636717348328995607258045, 8.219448215687862113054384715782, 9.045765223719250670382880151825, 9.961724949190421894828103529032, 11.34453549184167251032091221501

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