Properties

Label 2-483-161.61-c1-0-22
Degree $2$
Conductor $483$
Sign $0.832 - 0.554i$
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  + (1.66 + 1.30i)2-s + (−0.814 − 0.580i)3-s + (0.586 + 2.41i)4-s + (2.07 − 0.399i)5-s + (−0.596 − 2.03i)6-s + (−0.0453 − 2.64i)7-s + (−0.428 + 0.937i)8-s + (0.327 + 0.945i)9-s + (3.97 + 2.04i)10-s + (2.42 + 3.07i)11-s + (0.924 − 2.30i)12-s + (−0.763 − 1.18i)13-s + (3.38 − 4.46i)14-s + (−1.91 − 0.876i)15-s + (2.47 − 1.27i)16-s + (−0.200 − 0.191i)17-s + ⋯
L(s)  = 1  + (1.17 + 0.925i)2-s + (−0.470 − 0.334i)3-s + (0.293 + 1.20i)4-s + (0.926 − 0.178i)5-s + (−0.243 − 0.829i)6-s + (−0.0171 − 0.999i)7-s + (−0.151 + 0.331i)8-s + (0.109 + 0.315i)9-s + (1.25 + 0.647i)10-s + (0.730 + 0.928i)11-s + (0.266 − 0.666i)12-s + (−0.211 − 0.329i)13-s + (0.905 − 1.19i)14-s + (−0.495 − 0.226i)15-s + (0.619 − 0.319i)16-s + (−0.0486 − 0.0464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53392 + 0.767205i\)
\(L(\frac12)\) \(\approx\) \(2.53392 + 0.767205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (0.0453 + 2.64i)T \)
23 \( 1 + (-2.38 - 4.15i)T \)
good2 \( 1 + (-1.66 - 1.30i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-2.07 + 0.399i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-2.42 - 3.07i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (0.763 + 1.18i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.200 + 0.191i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-0.417 + 0.397i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (5.20 - 1.52i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.233 + 2.44i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (3.04 - 1.05i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (4.70 - 4.07i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.42 - 1.56i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-1.68 - 0.971i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.34 + 0.159i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (-3.02 + 5.86i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (4.91 + 6.90i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-3.71 - 9.28i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-1.43 + 10.0i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.40 + 0.583i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (9.79 - 0.466i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-8.30 + 9.58i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (12.8 + 1.22i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-11.3 - 13.0i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.27574302976485142174468674166, −10.09295388116466893727022581986, −9.451672987510876205314221787059, −7.79125620026098491625734274136, −7.06443711247259911687894214965, −6.39293379365013946325040233356, −5.41506485377483212495226284533, −4.67431037885905159808900369317, −3.55283689153724489209188421111, −1.57061811225511440042490852173, 1.78998637764941082432997318984, 2.90806739193934858360336748481, 4.02289846066562389327699381858, 5.24374540756339425070761175205, 5.79688759921568284299881173163, 6.65368444529467347270927768750, 8.522874346128035604176487934040, 9.380265070493060105814209561069, 10.32564834414763108483045329335, 11.13774121013971768469519987364

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