Properties

Label 2-483-161.160-c1-0-5
Degree $2$
Conductor $483$
Sign $0.0582 - 0.998i$
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  − 2.69·2-s + i·3-s + 5.27·4-s − 2.58·5-s − 2.69i·6-s + (−0.551 − 2.58i)7-s − 8.81·8-s − 9-s + 6.97·10-s − 3.13i·11-s + 5.27i·12-s + 4.69i·13-s + (1.48 + 6.97i)14-s − 2.58i·15-s + 13.2·16-s + 6.97·17-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.577i·3-s + 2.63·4-s − 1.15·5-s − 1.10i·6-s + (−0.208 − 0.978i)7-s − 3.11·8-s − 0.333·9-s + 2.20·10-s − 0.946i·11-s + 1.52i·12-s + 1.30i·13-s + (0.397 + 1.86i)14-s − 0.668i·15-s + 3.30·16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.0582 - 0.998i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.0582 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247092 + 0.233084i\)
\(L(\frac12)\) \(\approx\) \(0.247092 + 0.233084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (0.551 + 2.58i)T \)
23 \( 1 + (1.27 - 4.62i)T \)
good2 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
11 \( 1 + 3.13iT - 11T^{2} \)
13 \( 1 - 4.69iT - 13T^{2} \)
17 \( 1 - 6.97T + 17T^{2} \)
19 \( 1 + 3.13T + 19T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 + 2.57iT - 31T^{2} \)
37 \( 1 - 2.90iT - 37T^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 - 0.785iT - 43T^{2} \)
47 \( 1 - 12.3iT - 47T^{2} \)
53 \( 1 - 7.52iT - 53T^{2} \)
59 \( 1 + 3.15iT - 59T^{2} \)
61 \( 1 - 8.22T + 61T^{2} \)
67 \( 1 - 7.76iT - 67T^{2} \)
71 \( 1 - 9.23T + 71T^{2} \)
73 \( 1 - 9.65iT - 73T^{2} \)
79 \( 1 - 1.50iT - 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 2.18T + 89T^{2} \)
97 \( 1 - 7.74T + 97T^{2} \)
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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

−11.16099900487565929821542171023, −10.06907832027414684039160330542, −9.543112409144089584741048907841, −8.429369146890732584577067356708, −7.88480468181792025482846787873, −7.06719979451556161976227849716, −6.03691231366831957335531630565, −4.11263280042823089515400698507, −3.11942131376799219835572476234, −1.09091186066105895264905118689, 0.46040183125985371346753871338, 2.14013534385528242164906253730, 3.28964389272066275679358297939, 5.49578084875635693962682932511, 6.64746784152234966053154502564, 7.51066975347638164129394991671, 8.180456897714614610164673000444, 8.662979868328481918081262225839, 9.912102722962730775476656588870, 10.47165332878437366895139007150

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