Properties

Label 2-483-161.32-c1-0-24
Degree $2$
Conductor $483$
Sign $0.989 + 0.144i$
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  + (0.821 + 0.783i)2-s + (0.981 − 0.189i)3-s + (−0.0339 − 0.711i)4-s + (−1.35 + 1.06i)5-s + (0.954 + 0.613i)6-s + (1.68 − 2.03i)7-s + (2.01 − 2.32i)8-s + (0.928 − 0.371i)9-s + (−1.94 − 0.185i)10-s + (1.67 − 1.59i)11-s + (−0.167 − 0.692i)12-s + (−0.181 − 0.397i)13-s + (2.98 − 0.350i)14-s + (−1.12 + 1.30i)15-s + (2.05 − 0.196i)16-s + (−0.319 + 0.164i)17-s + ⋯
L(s)  = 1  + (0.580 + 0.553i)2-s + (0.566 − 0.109i)3-s + (−0.0169 − 0.355i)4-s + (−0.605 + 0.476i)5-s + (0.389 + 0.250i)6-s + (0.638 − 0.769i)7-s + (0.712 − 0.822i)8-s + (0.309 − 0.123i)9-s + (−0.615 − 0.0587i)10-s + (0.505 − 0.482i)11-s + (−0.0484 − 0.199i)12-s + (−0.0503 − 0.110i)13-s + (0.796 − 0.0936i)14-s + (−0.291 + 0.336i)15-s + (0.514 − 0.0491i)16-s + (−0.0775 + 0.0399i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.989 + 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26566 - 0.164876i\)
\(L(\frac12)\) \(\approx\) \(2.26566 - 0.164876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (-1.68 + 2.03i)T \)
23 \( 1 + (-2.83 - 3.86i)T \)
good2 \( 1 + (-0.821 - 0.783i)T + (0.0951 + 1.99i)T^{2} \)
5 \( 1 + (1.35 - 1.06i)T + (1.17 - 4.85i)T^{2} \)
11 \( 1 + (-1.67 + 1.59i)T + (0.523 - 10.9i)T^{2} \)
13 \( 1 + (0.181 + 0.397i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.319 - 0.164i)T + (9.86 - 13.8i)T^{2} \)
19 \( 1 + (3.90 + 2.01i)T + (11.0 + 15.4i)T^{2} \)
29 \( 1 + (-8.72 - 5.60i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-0.951 - 2.74i)T + (-24.3 + 19.1i)T^{2} \)
37 \( 1 + (0.0425 - 0.0170i)T + (26.7 - 25.5i)T^{2} \)
41 \( 1 + (-0.876 - 6.09i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (6.63 + 7.66i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (2.87 + 4.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.35 - 3.30i)T + (-17.3 + 50.0i)T^{2} \)
59 \( 1 + (9.11 + 0.870i)T + (57.9 + 11.1i)T^{2} \)
61 \( 1 + (6.55 + 1.26i)T + (56.6 + 22.6i)T^{2} \)
67 \( 1 + (0.788 - 3.24i)T + (-59.5 - 30.7i)T^{2} \)
71 \( 1 + (8.98 - 2.63i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-0.543 - 11.4i)T + (-72.6 + 6.93i)T^{2} \)
79 \( 1 + (-2.92 + 4.10i)T + (-25.8 - 74.6i)T^{2} \)
83 \( 1 + (1.45 - 10.1i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (1.07 - 3.11i)T + (-69.9 - 55.0i)T^{2} \)
97 \( 1 + (-1.80 - 12.5i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.86759619370344007009589551583, −10.26462462827489395674654858035, −9.012834898103235751198132464751, −8.065504791069716734482258490919, −7.08987278870236110163210263091, −6.59132222970746846431928194622, −5.15346947309135684725239822092, −4.24014332503410366540121963501, −3.26499650486976840124524894924, −1.32596472464378630543624880785, 1.92195640649746479912374095825, 2.99140073630164717816435843812, 4.42839389972413303168395536244, 4.60259653967046918555209805485, 6.29108758646336146207556108583, 7.71159477324785396610411059901, 8.337418444693787353797904549829, 8.980857957646430882102500593228, 10.25504370058247754326287359624, 11.29027991248295635843458691017

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