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Properties

Label 2-483-69.68-c1-0-13
Degree $2$
Conductor $483$
Sign $-0.487 - 0.873i$
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  + 2.64i·2-s + (−1.65 − 0.499i)3-s − 4.97·4-s + 2.97·5-s + (1.32 − 4.37i)6-s − i·7-s − 7.85i·8-s + (2.50 + 1.65i)9-s + 7.86i·10-s + 5.69·11-s + (8.25 + 2.48i)12-s + 3.06·13-s + 2.64·14-s + (−4.93 − 1.48i)15-s + 10.8·16-s − 5.87·17-s + ⋯
L(s)  = 1  + 1.86i·2-s + (−0.957 − 0.288i)3-s − 2.48·4-s + 1.33·5-s + (0.539 − 1.78i)6-s − 0.377i·7-s − 2.77i·8-s + (0.833 + 0.552i)9-s + 2.48i·10-s + 1.71·11-s + (2.38 + 0.717i)12-s + 0.849·13-s + 0.705·14-s + (−1.27 − 0.384i)15-s + 2.70·16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.487 - 0.873i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622172 + 1.05942i\)
\(L(\frac12)\) \(\approx\) \(0.622172 + 1.05942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 + 0.499i)T \)
7 \( 1 + iT \)
23 \( 1 + (-3.33 + 3.44i)T \)
good2 \( 1 - 2.64iT - 2T^{2} \)
5 \( 1 - 2.97T + 5T^{2} \)
11 \( 1 - 5.69T + 11T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 - 0.0296iT - 19T^{2} \)
29 \( 1 - 8.39iT - 29T^{2} \)
31 \( 1 - 4.36T + 31T^{2} \)
37 \( 1 - 6.88iT - 37T^{2} \)
41 \( 1 + 7.13iT - 41T^{2} \)
43 \( 1 - 3.13iT - 43T^{2} \)
47 \( 1 - 0.363iT - 47T^{2} \)
53 \( 1 - 3.88T + 53T^{2} \)
59 \( 1 + 4.67iT - 59T^{2} \)
61 \( 1 - 15.0iT - 61T^{2} \)
67 \( 1 + 5.80iT - 67T^{2} \)
71 \( 1 + 6.51iT - 71T^{2} \)
73 \( 1 - 0.133T + 73T^{2} \)
79 \( 1 - 9.73iT - 79T^{2} \)
83 \( 1 + 0.210T + 83T^{2} \)
89 \( 1 + 2.19T + 89T^{2} \)
97 \( 1 + 2.81iT - 97T^{2} \)
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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

−11.14646936266775997282793475760, −10.14668019904242359424175591123, −9.149565009021807387250373295786, −8.599431247256856208814241156850, −7.03498174109608919404482586365, −6.56974800543198350218852412143, −6.08611591895999649938427372397, −5.04046282545966169275353209356, −4.15663070446361047511195548255, −1.27144071254719667809770254017, 1.15335851391613993605455710951, 2.17143503257158232196124214173, 3.75286522710222143614331542065, 4.61940439054127147099387086756, 5.80056790066092321734443603139, 6.53550102080889325167423551783, 8.772085728375842547775925731259, 9.372666134498209086002239710444, 9.889469869309137164502198211602, 10.91312650658539947348792760766

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