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Properties

Label 2-483-161.17-c1-0-2
Degree $2$
Conductor $483$
Sign $-0.948 - 0.317i$
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  + (0.774 − 0.0739i)2-s + (0.690 + 0.723i)3-s + (−1.37 + 0.264i)4-s + (−0.903 − 0.465i)5-s + (0.587 + 0.509i)6-s + (−2.62 − 0.293i)7-s + (−2.53 + 0.743i)8-s + (−0.0475 + 0.998i)9-s + (−0.733 − 0.293i)10-s + (−0.303 + 3.17i)11-s + (−1.13 − 0.809i)12-s + (−3.36 + 0.483i)13-s + (−2.05 − 0.0324i)14-s + (−0.286 − 0.974i)15-s + (0.685 − 0.274i)16-s + (−1.55 − 4.48i)17-s + ⋯
L(s)  = 1  + (0.547 − 0.0522i)2-s + (0.398 + 0.417i)3-s + (−0.685 + 0.132i)4-s + (−0.403 − 0.208i)5-s + (0.239 + 0.207i)6-s + (−0.993 − 0.110i)7-s + (−0.895 + 0.262i)8-s + (−0.0158 + 0.332i)9-s + (−0.231 − 0.0928i)10-s + (−0.0915 + 0.958i)11-s + (−0.328 − 0.233i)12-s + (−0.933 + 0.134i)13-s + (−0.549 − 0.00868i)14-s + (−0.0739 − 0.251i)15-s + (0.171 − 0.0685i)16-s + (−0.376 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.948 - 0.317i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.948 - 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0796044 + 0.488744i\)
\(L(\frac12)\) \(\approx\) \(0.0796044 + 0.488744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.690 - 0.723i)T \)
7 \( 1 + (2.62 + 0.293i)T \)
23 \( 1 + (1.31 - 4.61i)T \)
good2 \( 1 + (-0.774 + 0.0739i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (0.903 + 0.465i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (0.303 - 3.17i)T + (-10.8 - 2.08i)T^{2} \)
13 \( 1 + (3.36 - 0.483i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.55 + 4.48i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (1.60 - 4.63i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (1.41 - 1.63i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-8.75 - 2.12i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (3.27 + 0.155i)T + (36.8 + 3.51i)T^{2} \)
41 \( 1 + (1.25 + 1.95i)T + (-17.0 + 37.2i)T^{2} \)
43 \( 1 + (-1.80 + 6.15i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (5.17 - 2.98i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.87 - 4.93i)T + (-12.4 - 51.5i)T^{2} \)
59 \( 1 + (-5.38 + 13.4i)T + (-42.7 - 40.7i)T^{2} \)
61 \( 1 + (-5.10 - 4.86i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (6.60 - 4.70i)T + (21.9 - 63.3i)T^{2} \)
71 \( 1 + (2.70 - 5.92i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-3.05 - 15.8i)T + (-67.7 + 27.1i)T^{2} \)
79 \( 1 + (-9.03 - 11.4i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (13.0 + 8.39i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (0.249 + 1.02i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (-10.8 + 6.99i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.72637068200141235114952411251, −10.01917021459118644531052765871, −9.801358509964207391292287700657, −8.819024743094519066767249378933, −7.82365761844851090140689798102, −6.81006390098235978445330527783, −5.44427693136648142944514246277, −4.51443867910229425137561394180, −3.74479043017216135980101448683, −2.59448481760571340257942465765, 0.23082208977573797064771473253, 2.69721037798809241213574997240, 3.61752641494630145411371261841, 4.69006920177457782717973079638, 6.02560780386531230103449892333, 6.64379907323127991576286769671, 7.968889899761500840824346713510, 8.751430270407622799753885842616, 9.594317508198293018483372231000, 10.51079891800409330699646756510

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