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Properties

Label 2-462-21.17-c1-0-17
Degree $2$
Conductor $462$
Sign $0.0285 + 0.999i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (−0.866 + 0.5i)2-s + (0.599 + 1.62i)3-s + (0.499 − 0.866i)4-s + (−1.46 − 2.53i)5-s + (−1.33 − 1.10i)6-s + (−2.19 + 1.48i)7-s + 0.999i·8-s + (−2.28 + 1.94i)9-s + (2.53 + 1.46i)10-s + (−0.866 − 0.5i)11-s + (1.70 + 0.292i)12-s − 5.44i·13-s + (1.15 − 2.38i)14-s + (3.24 − 3.90i)15-s + (−0.5 − 0.866i)16-s + (2.65 − 4.60i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.346 + 0.938i)3-s + (0.249 − 0.433i)4-s + (−0.655 − 1.13i)5-s + (−0.543 − 0.452i)6-s + (−0.827 + 0.560i)7-s + 0.353i·8-s + (−0.760 + 0.649i)9-s + (0.802 + 0.463i)10-s + (−0.261 − 0.150i)11-s + (0.492 + 0.0845i)12-s − 1.51i·13-s + (0.308 − 0.636i)14-s + (0.838 − 1.00i)15-s + (−0.125 − 0.216i)16-s + (0.644 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0285 + 0.999i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0285 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303588 - 0.295055i\)
\(L(\frac12)\) \(\approx\) \(0.303588 - 0.295055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.599 - 1.62i)T \)
7 \( 1 + (2.19 - 1.48i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (1.46 + 2.53i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.44iT - 13T^{2} \)
17 \( 1 + (-2.65 + 4.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.544 - 0.314i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.90 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.31iT - 29T^{2} \)
31 \( 1 + (6.56 + 3.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.22 + 3.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.60T + 41T^{2} \)
43 \( 1 + 4.14T + 43T^{2} \)
47 \( 1 + (-4.41 - 7.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.33 + 3.65i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.117 - 0.203i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.60 - 3.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 - 3.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.31iT - 71T^{2} \)
73 \( 1 + (-11.8 - 6.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.47 + 7.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.02T + 83T^{2} \)
89 \( 1 + (0.987 + 1.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.2iT - 97T^{2} \)
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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

−10.55824494394317062980878301564, −9.612582236537089009446048612318, −9.191533743896652316952780940963, −8.111973045584938789179752714635, −7.69570978199451268265138571173, −5.81429775700850080012608839191, −5.28624351756672351024734622438, −3.95372579242331547639542609335, −2.75336953366462266025592892395, −0.29282822497922738607927736738, 1.81480771325699851544743152439, 3.14044791773588225046828911671, 3.92505776286557774316797820731, 6.22620315550656549405456169195, 6.91209841288781466223195294987, 7.50445411208434605853354751747, 8.483468369969548379524938497774, 9.474478472200953257992669574584, 10.50591118951285556403400957308, 11.13429980280483712858726315537

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