Properties

Label 2-483-161.66-c1-0-11
Degree $2$
Conductor $483$
Sign $0.377 - 0.925i$
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  + (−1.47 + 1.16i)2-s + (−0.814 + 0.580i)3-s + (0.362 − 1.49i)4-s + (0.431 + 0.0832i)5-s + (0.529 − 1.80i)6-s + (1.82 + 1.91i)7-s + (−0.361 − 0.790i)8-s + (0.327 − 0.945i)9-s + (−0.735 + 0.378i)10-s + (2.95 − 3.76i)11-s + (0.571 + 1.42i)12-s + (−0.0227 + 0.0354i)13-s + (−4.92 − 0.709i)14-s + (−0.399 + 0.182i)15-s + (4.18 + 2.15i)16-s + (4.38 − 4.18i)17-s + ⋯
L(s)  = 1  + (−1.04 + 0.822i)2-s + (−0.470 + 0.334i)3-s + (0.181 − 0.747i)4-s + (0.193 + 0.0372i)5-s + (0.216 − 0.736i)6-s + (0.689 + 0.723i)7-s + (−0.127 − 0.279i)8-s + (0.109 − 0.315i)9-s + (−0.232 + 0.119i)10-s + (0.891 − 1.13i)11-s + (0.164 + 0.412i)12-s + (−0.00632 + 0.00983i)13-s + (−1.31 − 0.189i)14-s + (−0.103 + 0.0471i)15-s + (1.04 + 0.539i)16-s + (1.06 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.377 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680084 + 0.457019i\)
\(L(\frac12)\) \(\approx\) \(0.680084 + 0.457019i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 - 0.580i)T \)
7 \( 1 + (-1.82 - 1.91i)T \)
23 \( 1 + (2.18 + 4.27i)T \)
good2 \( 1 + (1.47 - 1.16i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (-0.431 - 0.0832i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (-2.95 + 3.76i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (0.0227 - 0.0354i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (-4.38 + 4.18i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (-1.72 - 1.64i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (-8.33 - 2.44i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.0526 - 0.551i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (-4.97 - 1.72i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (-5.73 - 4.96i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-3.61 - 1.65i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (11.2 - 6.50i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (13.9 - 0.662i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (-1.41 - 2.74i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (-4.84 + 6.80i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (0.183 - 0.458i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (0.420 + 2.92i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-1.96 - 0.475i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (-4.99 - 0.238i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (-4.93 - 5.69i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-11.3 + 1.08i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (-3.83 + 4.43i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.06487915619420932438637429704, −9.881812681887204850192610092635, −9.358319090181788428945754682943, −8.358584766669716422561687653386, −7.80823531843108810759643862257, −6.39707364457336862608607587433, −5.95350230981952100895806226637, −4.71562714504122973504040072600, −3.17462362134081372747311258651, −1.03641872821568402176247203701, 1.12539899915974017431074636972, 1.99068223512266233154196636799, 3.81178899879130085999835996634, 5.08790005928657637283718671904, 6.27585575050472862881224245212, 7.53664637348751165212141743063, 8.066341632707521583060006186160, 9.369310417100884960122829703554, 9.981593774348775444363970833579, 10.69033646697185579359247159830

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