Properties

Label 2-483-161.4-c1-0-13
Degree $2$
Conductor $483$
Sign $0.831 - 0.554i$
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  + (0.466 − 0.655i)2-s + (−0.235 + 0.971i)3-s + (0.442 + 1.27i)4-s + (−0.00881 + 0.185i)5-s + (0.526 + 0.608i)6-s + (1.59 − 2.10i)7-s + (2.58 + 0.760i)8-s + (−0.888 − 0.458i)9-s + (0.117 + 0.0921i)10-s + (1.88 + 2.64i)11-s + (−1.34 + 0.128i)12-s + (−0.0720 + 0.501i)13-s + (−0.635 − 2.03i)14-s + (−0.177 − 0.0521i)15-s + (−0.420 + 0.330i)16-s + (−0.392 + 0.0756i)17-s + ⋯
L(s)  = 1  + (0.330 − 0.463i)2-s + (−0.136 + 0.561i)3-s + (0.221 + 0.639i)4-s + (−0.00394 + 0.0827i)5-s + (0.215 + 0.248i)6-s + (0.604 − 0.796i)7-s + (0.915 + 0.268i)8-s + (−0.296 − 0.152i)9-s + (0.0370 + 0.0291i)10-s + (0.567 + 0.796i)11-s + (−0.388 + 0.0371i)12-s + (−0.0199 + 0.139i)13-s + (−0.169 − 0.543i)14-s + (−0.0458 − 0.0134i)15-s + (−0.105 + 0.0826i)16-s + (−0.0951 + 0.0183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.831 - 0.554i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.831 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78851 + 0.541790i\)
\(L(\frac12)\) \(\approx\) \(1.78851 + 0.541790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (-1.59 + 2.10i)T \)
23 \( 1 + (1.16 - 4.65i)T \)
good2 \( 1 + (-0.466 + 0.655i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (0.00881 - 0.185i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (-1.88 - 2.64i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (0.0720 - 0.501i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (0.392 - 0.0756i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (2.40 + 0.462i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (-2.98 - 3.44i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-5.00 + 4.76i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (7.11 + 3.67i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (-2.79 - 1.79i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (3.30 - 0.971i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-0.525 - 0.909i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.870 + 0.348i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (4.59 + 3.61i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (2.47 + 10.2i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (-4.52 - 0.432i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (5.94 + 13.0i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.10 + 6.08i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-8.29 + 3.32i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (3.39 - 2.18i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (9.06 + 8.64i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (4.84 + 3.11i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.05877156172822521114083286889, −10.47075911171752796853296837586, −9.432802532040916349213732746087, −8.338846875604689736859101407994, −7.41823334510112643900072370100, −6.57000564884397776675205982258, −4.90643910384038116410747914768, −4.27110164589286334454232762511, −3.28703646807829782285772249657, −1.77768835520895965577014692736, 1.22942917736643814298919935488, 2.61018302808101252237537118136, 4.45448062414864172379687735082, 5.41254074036559015477901217532, 6.26016819761542789227311270616, 6.91955540676385893326377218712, 8.260199567022514970319067649613, 8.787557644238508602980181340115, 10.18566604037123376439338972541, 10.96076561591369267638276280686

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