Properties

Label 2-483-161.4-c1-0-19
Degree $2$
Conductor $483$
Sign $0.574 + 0.818i$
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  + (−1.09 + 1.53i)2-s + (−0.235 + 0.971i)3-s + (−0.502 − 1.45i)4-s + (−0.0927 + 1.94i)5-s + (−1.23 − 1.42i)6-s + (0.850 − 2.50i)7-s + (−0.837 − 0.245i)8-s + (−0.888 − 0.458i)9-s + (−2.88 − 2.26i)10-s + (−2.62 − 3.68i)11-s + (1.52 − 0.145i)12-s + (0.355 − 2.47i)13-s + (2.91 + 4.03i)14-s + (−1.87 − 0.549i)15-s + (3.70 − 2.91i)16-s + (−5.23 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.771 + 1.08i)2-s + (−0.136 + 0.561i)3-s + (−0.251 − 0.725i)4-s + (−0.0414 + 0.870i)5-s + (−0.502 − 0.580i)6-s + (0.321 − 0.946i)7-s + (−0.296 − 0.0869i)8-s + (−0.296 − 0.152i)9-s + (−0.911 − 0.716i)10-s + (−0.790 − 1.11i)11-s + (0.441 − 0.0421i)12-s + (0.0985 − 0.685i)13-s + (0.777 + 1.07i)14-s + (−0.482 − 0.141i)15-s + (0.926 − 0.728i)16-s + (−1.27 + 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140819 - 0.0732194i\)
\(L(\frac12)\) \(\approx\) \(0.140819 - 0.0732194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (-0.850 + 2.50i)T \)
23 \( 1 + (4.10 + 2.47i)T \)
good2 \( 1 + (1.09 - 1.53i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (0.0927 - 1.94i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (2.62 + 3.68i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (-0.355 + 2.47i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (5.23 - 1.00i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (5.79 + 1.11i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (-1.14 - 1.32i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (4.23 - 4.03i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (4.15 + 2.14i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (-6.12 - 3.93i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (2.62 - 0.769i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (2.16 + 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.68 - 1.47i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (-1.11 - 0.875i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (-2.31 - 9.53i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (-11.0 - 1.05i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (0.430 + 0.941i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.04 + 5.90i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (13.5 - 5.40i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (12.0 - 7.73i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-6.53 - 6.23i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (4.56 + 2.93i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.74015242811150745677764641840, −10.04803330865267162483751394802, −8.646544152807110787685174948148, −8.283705668487606718695644503761, −7.13397071329703324521728965843, −6.47571467717776427475687716612, −5.48737243952347377512306722374, −4.09333912889677023245791168274, −2.87229524517478311496033184099, −0.11713717661484256893620258460, 1.84022176568237344025793101313, 2.33367222310639978404894290405, 4.30079398601551678329376479675, 5.40068366104843068695328484793, 6.53848990640535342112774828038, 7.906999566761803172138260178888, 8.697807210749029143737230101665, 9.259867754803006303826024808708, 10.25067339321970184528282092382, 11.24646278478172715368897365569

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