Properties

Label 2-483-161.9-c1-0-1
Degree $2$
Conductor $483$
Sign $-0.433 + 0.901i$
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  + (−0.460 + 1.89i)2-s + (−0.327 + 0.945i)3-s + (−1.61 − 0.831i)4-s + (0.128 + 0.0513i)5-s + (−1.64 − 1.05i)6-s + (−1.62 + 2.09i)7-s + (−0.237 + 0.274i)8-s + (−0.786 − 0.618i)9-s + (−0.156 + 0.219i)10-s + (0.270 + 1.11i)11-s + (1.31 − 1.25i)12-s + (0.526 + 1.15i)13-s + (−3.22 − 4.03i)14-s + (−0.0905 + 0.104i)15-s + (−2.51 − 3.53i)16-s + (−0.146 + 3.08i)17-s + ⋯
L(s)  = 1  + (−0.325 + 1.34i)2-s + (−0.188 + 0.545i)3-s + (−0.806 − 0.415i)4-s + (0.0573 + 0.0229i)5-s + (−0.670 − 0.431i)6-s + (−0.612 + 0.790i)7-s + (−0.0840 + 0.0970i)8-s + (−0.262 − 0.206i)9-s + (−0.0495 + 0.0695i)10-s + (0.0814 + 0.335i)11-s + (0.378 − 0.361i)12-s + (0.146 + 0.319i)13-s + (−0.861 − 1.07i)14-s + (−0.0233 + 0.0269i)15-s + (−0.628 − 0.883i)16-s + (−0.0355 + 0.747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344303 - 0.547844i\)
\(L(\frac12)\) \(\approx\) \(0.344303 - 0.547844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 - 0.945i)T \)
7 \( 1 + (1.62 - 2.09i)T \)
23 \( 1 + (4.72 - 0.813i)T \)
good2 \( 1 + (0.460 - 1.89i)T + (-1.77 - 0.916i)T^{2} \)
5 \( 1 + (-0.128 - 0.0513i)T + (3.61 + 3.45i)T^{2} \)
11 \( 1 + (-0.270 - 1.11i)T + (-9.77 + 5.04i)T^{2} \)
13 \( 1 + (-0.526 - 1.15i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.146 - 3.08i)T + (-16.9 - 1.61i)T^{2} \)
19 \( 1 + (0.226 + 4.75i)T + (-18.9 + 1.80i)T^{2} \)
29 \( 1 + (-4.44 - 2.85i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (1.86 + 0.358i)T + (28.7 + 11.5i)T^{2} \)
37 \( 1 + (4.39 + 3.45i)T + (8.72 + 35.9i)T^{2} \)
41 \( 1 + (-0.573 - 3.98i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-4.23 - 4.88i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (1.76 - 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.2 + 0.981i)T + (52.0 - 10.0i)T^{2} \)
59 \( 1 + (5.01 - 7.04i)T + (-19.2 - 55.7i)T^{2} \)
61 \( 1 + (-2.21 - 6.40i)T + (-47.9 + 37.7i)T^{2} \)
67 \( 1 + (2.99 + 2.85i)T + (3.18 + 66.9i)T^{2} \)
71 \( 1 + (2.58 - 0.759i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-10.2 - 5.27i)T + (42.3 + 59.4i)T^{2} \)
79 \( 1 + (10.6 + 1.01i)T + (77.5 + 14.9i)T^{2} \)
83 \( 1 + (1.61 - 11.2i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-3.25 + 0.627i)T + (82.6 - 33.0i)T^{2} \)
97 \( 1 + (-2.43 - 16.9i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.61890657545728421289181260872, −10.43905059877461109263546960691, −9.472348831870469724012379636866, −8.863670329576120661451974366450, −8.006667925668272568931371921341, −6.82873621038971292360198667040, −6.14804568301225337722663528383, −5.34653929773202985682962548451, −4.15309497783652008162790170400, −2.56538061242525460717385335097, 0.43130950607139581813754405967, 1.81381630161369058246029464514, 3.15898099469178838440760669944, 4.05504295944674992375953961667, 5.72997407142528833363105566026, 6.69915910370643437011475255968, 7.75144416471109314977968341235, 8.813971550845017193514514946160, 9.899918480686621363585765409887, 10.34526460411842825656807887887

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