Properties

Label 2-483-23.13-c1-0-18
Degree $2$
Conductor $483$
Sign $-0.276 + 0.960i$
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  + (−0.544 − 0.627i)2-s + (0.841 + 0.540i)3-s + (0.186 − 1.29i)4-s + (0.405 − 0.888i)5-s + (−0.118 − 0.822i)6-s + (−0.959 − 0.281i)7-s + (−2.31 + 1.48i)8-s + (0.415 + 0.909i)9-s + (−0.778 + 0.228i)10-s + (3.57 − 4.12i)11-s + (0.857 − 0.989i)12-s + (−0.954 + 0.280i)13-s + (0.345 + 0.755i)14-s + (0.821 − 0.528i)15-s + (−0.321 − 0.0943i)16-s + (−0.595 − 4.14i)17-s + ⋯
L(s)  = 1  + (−0.384 − 0.443i)2-s + (0.485 + 0.312i)3-s + (0.0931 − 0.648i)4-s + (0.181 − 0.397i)5-s + (−0.0482 − 0.335i)6-s + (−0.362 − 0.106i)7-s + (−0.817 + 0.525i)8-s + (0.138 + 0.303i)9-s + (−0.246 + 0.0723i)10-s + (1.07 − 1.24i)11-s + (0.247 − 0.285i)12-s + (−0.264 + 0.0777i)13-s + (0.0922 + 0.201i)14-s + (0.212 − 0.136i)15-s + (−0.0803 − 0.0235i)16-s + (−0.144 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.276 + 0.960i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.774710 - 1.02910i\)
\(L(\frac12)\) \(\approx\) \(0.774710 - 1.02910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (1.96 + 4.37i)T \)
good2 \( 1 + (0.544 + 0.627i)T + (-0.284 + 1.97i)T^{2} \)
5 \( 1 + (-0.405 + 0.888i)T + (-3.27 - 3.77i)T^{2} \)
11 \( 1 + (-3.57 + 4.12i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.954 - 0.280i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.595 + 4.14i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.0835 + 0.581i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.113 + 0.792i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (3.31 - 2.12i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (1.49 + 3.28i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-4.02 + 8.80i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-9.13 - 5.87i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + (-0.848 - 0.249i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-6.25 + 1.83i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-1.25 + 0.807i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (0.874 + 1.00i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-7.26 - 8.38i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.33 - 9.29i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-8.76 + 2.57i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (1.31 + 2.88i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-3.21 - 2.06i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (7.38 - 16.1i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.72769336262258332611656145417, −9.611036795887033853543965447883, −9.166612710752688851299780173246, −8.465513919321192497211309631864, −6.99998247611213965654654418906, −6.01103029940540705360108318093, −4.99603276602422769800945278112, −3.64185810957313790064599270908, −2.42441427229982124991963843290, −0.863739051355652633961554653849, 1.97283003788607766352862240645, 3.29476258295762986240427555651, 4.28207979043574461945283389644, 6.11151182241193724617853981886, 6.83344260368002425162265692390, 7.55893175810140674782678473645, 8.473460097337733271882313137336, 9.392720364466887880297683975646, 9.974474474486547469160679991542, 11.34451294871062064431443631310

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