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Properties

Label 2-483-161.10-c1-0-22
Degree $2$
Conductor $483$
Sign $0.276 + 0.961i$
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  + (0.369 − 1.06i)2-s + (−0.458 − 0.888i)3-s + (0.568 + 0.447i)4-s + (3.97 − 0.379i)5-s + (−1.11 + 0.160i)6-s + (−2.62 + 0.319i)7-s + (2.58 − 1.66i)8-s + (−0.580 + 0.814i)9-s + (1.06 − 4.38i)10-s + (0.579 − 0.200i)11-s + (0.136 − 0.710i)12-s + (−0.0130 − 0.0444i)13-s + (−0.629 + 2.92i)14-s + (−2.15 − 3.35i)15-s + (−0.478 − 1.97i)16-s + (2.93 + 1.17i)17-s + ⋯
L(s)  = 1  + (0.261 − 0.754i)2-s + (−0.264 − 0.513i)3-s + (0.284 + 0.223i)4-s + (1.77 − 0.169i)5-s + (−0.456 + 0.0656i)6-s + (−0.992 + 0.120i)7-s + (0.915 − 0.588i)8-s + (−0.193 + 0.271i)9-s + (0.336 − 1.38i)10-s + (0.174 − 0.0605i)11-s + (0.0395 − 0.205i)12-s + (−0.00362 − 0.0123i)13-s + (−0.168 + 0.780i)14-s + (−0.557 − 0.867i)15-s + (−0.119 − 0.492i)16-s + (0.710 + 0.284i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.276 + 0.961i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67090 - 1.25834i\)
\(L(\frac12)\) \(\approx\) \(1.67090 - 1.25834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (2.62 - 0.319i)T \)
23 \( 1 + (3.88 - 2.80i)T \)
good2 \( 1 + (-0.369 + 1.06i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-3.97 + 0.379i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-0.579 + 0.200i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.0130 + 0.0444i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-2.93 - 1.17i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-0.106 + 0.0425i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.704 + 4.90i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.899 - 0.0428i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (6.97 + 4.96i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (2.62 + 1.20i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (3.51 - 5.46i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (11.8 - 6.82i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.16 + 6.46i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-5.96 - 1.44i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-3.58 - 1.84i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.38 - 7.20i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (2.10 + 2.43i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.180 + 0.229i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-7.96 + 8.35i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-5.02 - 10.9i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.699 - 14.6i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (1.42 - 3.12i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.76909222620848279744894782292, −9.929794575624138550215119667370, −9.484906799334345580817651292905, −8.076547786967357573236354911390, −6.80573496680740257680850364517, −6.17819007325757224581801756011, −5.27298883201076631168545654259, −3.58520963061875797141001614838, −2.44593609280144413915208760203, −1.50165371032904947260584125491, 1.81182063434162219828666151227, 3.22325868991957468648352987666, 4.95692941289398584939478036874, 5.66993075487572932407504285137, 6.43602195240652378849069449202, 6.98426899454954204921629120383, 8.564020798468720417678550040843, 9.735498135493861811652113309198, 10.06818189209672941492687151322, 10.81845113879754921164929333455

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