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Properties

Label 2-483-69.68-c1-0-0
Degree $2$
Conductor $483$
Sign $-0.510 + 0.859i$
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  + 1.98i·2-s + (0.949 − 1.44i)3-s − 1.92·4-s − 4.16·5-s + (2.87 + 1.88i)6-s + i·7-s + 0.139i·8-s + (−1.19 − 2.75i)9-s − 8.25i·10-s + 0.579·11-s + (−1.83 + 2.79i)12-s − 3.14·13-s − 1.98·14-s + (−3.95 + 6.03i)15-s − 4.13·16-s − 6.11·17-s + ⋯
L(s)  = 1  + 1.40i·2-s + (0.548 − 0.836i)3-s − 0.964·4-s − 1.86·5-s + (1.17 + 0.768i)6-s + 0.377i·7-s + 0.0494i·8-s + (−0.398 − 0.917i)9-s − 2.61i·10-s + 0.174·11-s + (−0.529 + 0.806i)12-s − 0.872·13-s − 0.529·14-s + (−1.02 + 1.55i)15-s − 1.03·16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.510 + 0.859i$
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: \((2,\ 483,\ (\ :1/2),\ -0.510 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0312618 - 0.0549087i\)
\(L(\frac12)\) \(\approx\) \(0.0312618 - 0.0549087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.949 + 1.44i)T \)
7 \( 1 - iT \)
23 \( 1 + (0.215 + 4.79i)T \)
good2 \( 1 - 1.98iT - 2T^{2} \)
5 \( 1 + 4.16T + 5T^{2} \)
11 \( 1 - 0.579T + 11T^{2} \)
13 \( 1 + 3.14T + 13T^{2} \)
17 \( 1 + 6.11T + 17T^{2} \)
19 \( 1 + 1.67iT - 19T^{2} \)
29 \( 1 - 4.32iT - 29T^{2} \)
31 \( 1 + 9.63T + 31T^{2} \)
37 \( 1 - 5.13iT - 37T^{2} \)
41 \( 1 - 6.82iT - 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 - 9.21T + 53T^{2} \)
59 \( 1 + 0.0193iT - 59T^{2} \)
61 \( 1 - 8.32iT - 61T^{2} \)
67 \( 1 - 4.78iT - 67T^{2} \)
71 \( 1 + 1.09iT - 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 + 1.77iT - 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + 6.78T + 89T^{2} \)
97 \( 1 + 7.76iT - 97T^{2} \)
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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

−11.74524726458345344133023186187, −10.94351019453239591043562446674, −8.979792233165150298767947914415, −8.642167287914560665059890668670, −7.73290212444743443878598785281, −7.11118116168949940360453185473, −6.55626261961944481720427616954, −5.05886177283814757586070337276, −4.03118292546314453964017109573, −2.57518247984913941579223711080, 0.03375384287773553190014024664, 2.29429579868218026518368219653, 3.61422989814449232985267369672, 3.97210956527849784653157283909, 4.90760982584801590608169917381, 7.07037550145588604398439247731, 7.85035083715125111421184515803, 8.911021697742372821538926625104, 9.621121084171225983230845114554, 10.74215146302692881363843660909

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