Properties

Label 2-483-161.16-c1-0-0
Degree $2$
Conductor $483$
Sign $0.676 - 0.736i$
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  + (−0.620 − 1.79i)2-s + (−0.888 − 0.458i)3-s + (−1.26 + 0.991i)4-s + (−3.16 − 0.302i)5-s + (−0.270 + 1.87i)6-s + (1.40 − 2.24i)7-s + (−0.632 − 0.406i)8-s + (0.580 + 0.814i)9-s + (1.42 + 5.87i)10-s + (−0.842 + 2.43i)11-s + (1.57 − 0.303i)12-s + (−3.50 − 1.02i)13-s + (−4.89 − 1.11i)14-s + (2.67 + 1.72i)15-s + (−1.09 + 4.50i)16-s + (−1.43 + 0.574i)17-s + ⋯
L(s)  = 1  + (−0.439 − 1.26i)2-s + (−0.513 − 0.264i)3-s + (−0.630 + 0.495i)4-s + (−1.41 − 0.135i)5-s + (−0.110 + 0.767i)6-s + (0.529 − 0.848i)7-s + (−0.223 − 0.143i)8-s + (0.193 + 0.271i)9-s + (0.450 + 1.85i)10-s + (−0.254 + 0.734i)11-s + (0.454 − 0.0876i)12-s + (−0.970 − 0.285i)13-s + (−1.30 − 0.298i)14-s + (0.691 + 0.444i)15-s + (−0.273 + 1.12i)16-s + (−0.347 + 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.676 - 0.736i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.676 - 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0275605 + 0.0121109i\)
\(L(\frac12)\) \(\approx\) \(0.0275605 + 0.0121109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.888 + 0.458i)T \)
7 \( 1 + (-1.40 + 2.24i)T \)
23 \( 1 + (-2.95 + 3.77i)T \)
good2 \( 1 + (0.620 + 1.79i)T + (-1.57 + 1.23i)T^{2} \)
5 \( 1 + (3.16 + 0.302i)T + (4.90 + 0.946i)T^{2} \)
11 \( 1 + (0.842 - 2.43i)T + (-8.64 - 6.79i)T^{2} \)
13 \( 1 + (3.50 + 1.02i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.43 - 0.574i)T + (12.3 - 11.7i)T^{2} \)
19 \( 1 + (-2.71 - 1.08i)T + (13.7 + 13.1i)T^{2} \)
29 \( 1 + (1.35 - 9.39i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (0.0236 - 0.497i)T + (-30.8 - 2.94i)T^{2} \)
37 \( 1 + (-4.33 - 6.08i)T + (-12.1 + 34.9i)T^{2} \)
41 \( 1 + (3.04 + 6.66i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.95 + 1.90i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (4.58 - 7.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.25 + 5.01i)T + (2.52 + 52.9i)T^{2} \)
59 \( 1 + (0.828 + 3.41i)T + (-52.4 + 27.0i)T^{2} \)
61 \( 1 + (8.94 - 4.61i)T + (35.3 - 49.6i)T^{2} \)
67 \( 1 + (4.79 + 0.924i)T + (62.2 + 24.9i)T^{2} \)
71 \( 1 + (0.887 - 1.02i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (6.13 - 4.82i)T + (17.2 - 70.9i)T^{2} \)
79 \( 1 + (5.24 - 5.00i)T + (3.75 - 78.9i)T^{2} \)
83 \( 1 + (0.967 - 2.11i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-0.309 - 6.49i)T + (-88.5 + 8.45i)T^{2} \)
97 \( 1 + (6.82 + 14.9i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.03439406681016985928219084266, −10.58037938250127887959311227364, −9.637206011822944718243774313298, −8.430489353816393874511309227392, −7.53467299595756930417174819154, −6.86023896590318317985050378367, −4.99156898265289884903170215194, −4.19309518096601932011764994837, −2.98325596877707234206080974649, −1.37005876036602198327076222108, 0.02384123715792150667715563545, 2.91709668508307805146275761243, 4.46455396262396143452235864821, 5.36739080701746643801579670279, 6.29052567316942943369244347089, 7.49266157491285824589752031803, 7.82824454540406963338096669486, 8.851683304494790841259903497203, 9.624069096864720491744947811065, 11.19953292579725291396068381539

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