Properties

Label 2-483-23.8-c1-0-17
Degree $2$
Conductor $483$
Sign $0.285 + 0.958i$
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  + (0.186 − 1.29i)2-s + (0.415 + 0.909i)3-s + (0.273 + 0.0801i)4-s + (−0.810 − 0.935i)5-s + (1.25 − 0.368i)6-s + (0.841 + 0.540i)7-s + (1.24 − 2.72i)8-s + (−0.654 + 0.755i)9-s + (−1.36 + 0.876i)10-s + (−0.722 − 5.02i)11-s + (0.0405 + 0.281i)12-s + (−1.31 + 0.847i)13-s + (0.857 − 0.989i)14-s + (0.514 − 1.12i)15-s + (−2.81 − 1.81i)16-s + (5.77 − 1.69i)17-s + ⋯
L(s)  = 1  + (0.131 − 0.916i)2-s + (0.239 + 0.525i)3-s + (0.136 + 0.0400i)4-s + (−0.362 − 0.418i)5-s + (0.513 − 0.150i)6-s + (0.317 + 0.204i)7-s + (0.439 − 0.962i)8-s + (−0.218 + 0.251i)9-s + (−0.431 + 0.277i)10-s + (−0.217 − 1.51i)11-s + (0.0116 + 0.0813i)12-s + (−0.365 + 0.234i)13-s + (0.229 − 0.264i)14-s + (0.132 − 0.290i)15-s + (−0.704 − 0.452i)16-s + (1.40 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.285 + 0.958i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.285 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46450 - 1.09150i\)
\(L(\frac12)\) \(\approx\) \(1.46450 - 1.09150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (-2.46 - 4.11i)T \)
good2 \( 1 + (-0.186 + 1.29i)T + (-1.91 - 0.563i)T^{2} \)
5 \( 1 + (0.810 + 0.935i)T + (-0.711 + 4.94i)T^{2} \)
11 \( 1 + (0.722 + 5.02i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (1.31 - 0.847i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-5.77 + 1.69i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-5.51 - 1.62i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (4.19 - 1.23i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-1.03 + 2.26i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (5.56 - 6.42i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (-3.91 - 4.52i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (0.842 + 1.84i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 + (5.44 + 3.50i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (10.4 - 6.68i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (1.36 - 2.99i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (1.72 - 12.0i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-1.16 + 8.11i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (4.18 + 1.22i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (4.51 - 2.90i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-2.21 + 2.55i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-2.88 - 6.32i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-11.2 - 12.9i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.94650428354336627018220547202, −10.02091214295964958856198510537, −9.263946372940246651945175169177, −8.149368463946179921421317026132, −7.42123322476718895734006145452, −5.84982755258474423939690241947, −4.88280893398359923471823176228, −3.54809700264256134753727696716, −2.96429627631466122672725332370, −1.20243266808519184061687419945, 1.77267774594665508376739593286, 3.15599825428793735001559057885, 4.76735865594058311199842997648, 5.63756153911153426301438288397, 6.89667839268734161444614010312, 7.47835186748047486869327590783, 7.86186976073327197732062409215, 9.245812792296992862828877152793, 10.30921029574937200601073818373, 11.17857117482997423949702816169

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