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Properties

Label 2-483-161.18-c1-0-14
Degree $2$
Conductor $483$
Sign $0.865 - 0.501i$
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  + (0.0498 + 0.205i)2-s + (0.327 + 0.945i)3-s + (1.73 − 0.895i)4-s + (−0.0996 + 0.0398i)5-s + (−0.177 + 0.114i)6-s + (2.55 − 0.688i)7-s + (0.547 + 0.632i)8-s + (−0.786 + 0.618i)9-s + (−0.0131 − 0.0184i)10-s + (−1.15 + 4.74i)11-s + (1.41 + 1.34i)12-s + (−0.0477 + 0.104i)13-s + (0.268 + 0.490i)14-s + (−0.0702 − 0.0810i)15-s + (2.16 − 3.04i)16-s + (−0.0660 − 1.38i)17-s + ⋯
L(s)  = 1  + (0.0352 + 0.145i)2-s + (0.188 + 0.545i)3-s + (0.868 − 0.447i)4-s + (−0.0445 + 0.0178i)5-s + (−0.0726 + 0.0466i)6-s + (0.965 − 0.260i)7-s + (0.193 + 0.223i)8-s + (−0.262 + 0.206i)9-s + (−0.00416 − 0.00584i)10-s + (−0.347 + 1.43i)11-s + (0.408 + 0.389i)12-s + (−0.0132 + 0.0289i)13-s + (0.0718 + 0.131i)14-s + (−0.0181 − 0.0209i)15-s + (0.541 − 0.760i)16-s + (−0.0160 − 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.865 - 0.501i$
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: \((2,\ 483,\ (\ :1/2),\ 0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92035 + 0.516651i\)
\(L(\frac12)\) \(\approx\) \(1.92035 + 0.516651i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 + 0.688i)T \)
23 \( 1 + (-4.57 + 1.42i)T \)
good2 \( 1 + (-0.0498 - 0.205i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (0.0996 - 0.0398i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (1.15 - 4.74i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (0.0477 - 0.104i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.0660 + 1.38i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.228 + 4.79i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-0.414 + 0.266i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-3.44 + 0.663i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (0.180 - 0.141i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (1.28 - 8.92i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (7.93 - 9.16i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (2.99 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.73 + 0.356i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (7.56 + 10.6i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (-2.82 + 8.17i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (5.02 - 4.78i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-4.61 - 1.35i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-4.39 + 2.26i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (7.40 - 0.707i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (0.803 + 5.58i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (10.6 + 2.04i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-0.454 + 3.16i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.23941290364860550572097676431, −10.11999527012980583506141613560, −9.537884946382584292498382247814, −8.199557005681531611761726438080, −7.38878812226900445589108708146, −6.57770435937227873746585975133, −5.07825342626598903513047953982, −4.66574753447797901834895786949, −2.91129698557849384168381237325, −1.70471455706607085883605007356, 1.48343692417445564928772697250, 2.71322098961966650791234347954, 3.80011915791801440142559905574, 5.44975514974165960970063813448, 6.27830799910938154788110145708, 7.42824437862598236527539084025, 8.170615836695187505331838875696, 8.719874322106653754162844873470, 10.34455796071654332815734946615, 11.04247887422836640594788151551

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