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Properties

Label 2-483-161.18-c1-0-31
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: $\chi_{483} (340, \cdot )$
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

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

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