Properties

Label 2-483-161.18-c1-0-13
Degree $2$
Conductor $483$
Sign $-0.558 - 0.829i$
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  + (0.490 + 2.02i)2-s + (0.327 + 0.945i)3-s + (−2.07 + 1.07i)4-s + (2.66 − 1.06i)5-s + (−1.75 + 1.12i)6-s + (0.754 − 2.53i)7-s + (−0.457 − 0.527i)8-s + (−0.786 + 0.618i)9-s + (3.46 + 4.87i)10-s + (−0.258 + 1.06i)11-s + (−1.69 − 1.61i)12-s + (−1.08 + 2.36i)13-s + (5.50 + 0.282i)14-s + (1.88 + 2.17i)15-s + (−1.86 + 2.62i)16-s + (0.237 + 4.98i)17-s + ⋯
L(s)  = 1  + (0.347 + 1.43i)2-s + (0.188 + 0.545i)3-s + (−1.03 + 0.535i)4-s + (1.19 − 0.477i)5-s + (−0.715 + 0.459i)6-s + (0.285 − 0.958i)7-s + (−0.161 − 0.186i)8-s + (−0.262 + 0.206i)9-s + (1.09 + 1.54i)10-s + (−0.0780 + 0.321i)11-s + (−0.487 − 0.465i)12-s + (−0.299 + 0.656i)13-s + (1.47 + 0.0754i)14-s + (0.485 + 0.560i)15-s + (−0.466 + 0.655i)16-s + (0.0576 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01406 + 1.90496i\)
\(L(\frac12)\) \(\approx\) \(1.01406 + 1.90496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (-0.754 + 2.53i)T \)
23 \( 1 + (-2.89 + 3.82i)T \)
good2 \( 1 + (-0.490 - 2.02i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (-2.66 + 1.06i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (0.258 - 1.06i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (1.08 - 2.36i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.237 - 4.98i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (0.00132 - 0.0279i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-3.29 + 2.11i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.56 - 0.879i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (-3.72 + 2.93i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (-1.66 + 11.5i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-1.11 + 1.28i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (0.953 + 1.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.35 + 0.798i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (-0.779 - 1.09i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (2.00 - 5.80i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (-4.78 + 4.56i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (0.798 + 0.234i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-4.00 + 2.06i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (-2.29 + 0.219i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (2.50 + 17.3i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (14.9 + 2.88i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (2.13 - 14.8i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.97212748527960765061010120686, −10.26977872669869616487975703285, −9.285760412065108669854536059047, −8.518037931775013212883572013866, −7.50415544371761093256029517578, −6.59465284638311184638999032382, −5.70130829458191811934566655533, −4.80841405346971210523496215645, −4.03322188310230801930851129984, −1.93430962986498003558635397726, 1.40671494228176027480533862195, 2.59306253201542426062725689110, 3.03917121094670372454688481877, 4.91941653576841738283877240029, 5.75090469003896814044075713059, 6.89023852687290651298757036873, 8.122224238136593983823363252921, 9.481237472693784579635777709557, 9.682241493083869106954631168120, 10.98877988931130511693747367922

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