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Properties

Label 2-483-161.25-c1-0-22
Degree $2$
Conductor $483$
Sign $-0.982 + 0.183i$
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  + (0.0924 − 1.94i)2-s + (−0.928 − 0.371i)3-s + (−1.76 − 0.168i)4-s + (0.940 + 3.87i)5-s + (−0.807 + 1.76i)6-s + (−1.58 − 2.11i)7-s + (0.0623 − 0.433i)8-s + (0.723 + 0.690i)9-s + (7.60 − 1.46i)10-s + (−0.161 − 3.39i)11-s + (1.57 + 0.813i)12-s + (−2.43 − 2.80i)13-s + (−4.25 + 2.88i)14-s + (0.567 − 3.94i)15-s + (−4.32 − 0.832i)16-s + (−1.66 − 2.34i)17-s + ⋯
L(s)  = 1  + (0.0653 − 1.37i)2-s + (−0.535 − 0.214i)3-s + (−0.883 − 0.0843i)4-s + (0.420 + 1.73i)5-s + (−0.329 + 0.721i)6-s + (−0.600 − 0.799i)7-s + (0.0220 − 0.153i)8-s + (0.241 + 0.230i)9-s + (2.40 − 0.463i)10-s + (−0.0487 − 1.02i)11-s + (0.455 + 0.234i)12-s + (−0.674 − 0.777i)13-s + (−1.13 + 0.771i)14-s + (0.146 − 1.01i)15-s + (−1.08 − 0.208i)16-s + (−0.404 − 0.567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.982 + 0.183i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.982 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0907981 - 0.979249i\)
\(L(\frac12)\) \(\approx\) \(0.0907981 - 0.979249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (1.58 + 2.11i)T \)
23 \( 1 + (-0.0390 + 4.79i)T \)
good2 \( 1 + (-0.0924 + 1.94i)T + (-1.99 - 0.190i)T^{2} \)
5 \( 1 + (-0.940 - 3.87i)T + (-4.44 + 2.29i)T^{2} \)
11 \( 1 + (0.161 + 3.39i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (2.43 + 2.80i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (1.66 + 2.34i)T + (-5.56 + 16.0i)T^{2} \)
19 \( 1 + (-2.81 + 3.95i)T + (-6.21 - 17.9i)T^{2} \)
29 \( 1 + (-3.14 + 6.88i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-5.49 - 4.31i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (4.04 + 3.85i)T + (1.76 + 36.9i)T^{2} \)
41 \( 1 + (-8.04 - 2.36i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.792 - 5.51i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (-1.81 - 3.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.16 - 3.36i)T + (-41.6 + 32.7i)T^{2} \)
59 \( 1 + (-7.25 + 1.39i)T + (54.7 - 21.9i)T^{2} \)
61 \( 1 + (10.0 - 4.03i)T + (44.1 - 42.0i)T^{2} \)
67 \( 1 + (6.11 - 3.15i)T + (38.8 - 54.5i)T^{2} \)
71 \( 1 + (1.06 + 0.684i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (10.8 + 1.03i)T + (71.6 + 13.8i)T^{2} \)
79 \( 1 + (-1.62 + 4.68i)T + (-62.0 - 48.8i)T^{2} \)
83 \( 1 + (-4.29 + 1.26i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-3.11 + 2.44i)T + (20.9 - 86.4i)T^{2} \)
97 \( 1 + (-10.1 - 2.98i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.55952945726442430567161918872, −10.27476114229929602295985342216, −9.336381175966479156930795987558, −7.57204761700046660914340730803, −6.79276858657387956734355568529, −6.04517206936340382255463320299, −4.42947976325471285918044323038, −3.00539474302339459165830888245, −2.70673397278730621468226165242, −0.61324971081417830872249843466, 1.86766319888788854637092691014, 4.30257199786161664828725783297, 5.12667591689927481084790252448, 5.69883985574550243396353301152, 6.63716916347499237487931505260, 7.68619394801927566079402549586, 8.755765808702366235458180362537, 9.313730947882259624416102431601, 10.09708607013622668048412663342, 11.83869140567489416256444676657

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