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Properties

Label 2-483-161.19-c1-0-31
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: $\chi_{483} (19, \cdot )$
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

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

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