Properties

Label 2-84-84.11-c1-0-3
Degree $2$
Conductor $84$
Sign $0.534 - 0.844i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
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  + (1.29 + 0.561i)2-s + (−1.27 + 1.16i)3-s + (1.36 + 1.45i)4-s + (−0.432 + 0.249i)5-s + (−2.31 + 0.798i)6-s + (−0.261 − 2.63i)7-s + (0.956 + 2.66i)8-s + (0.267 − 2.98i)9-s + (−0.701 + 0.0810i)10-s + (0.695 − 1.20i)11-s + (−3.45 − 0.264i)12-s + 2.75·13-s + (1.13 − 3.56i)14-s + (0.260 − 0.824i)15-s + (−0.255 + 3.99i)16-s + (−5.04 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.917 + 0.397i)2-s + (−0.737 + 0.674i)3-s + (0.684 + 0.729i)4-s + (−0.193 + 0.111i)5-s + (−0.945 + 0.326i)6-s + (−0.0989 − 0.995i)7-s + (0.338 + 0.941i)8-s + (0.0890 − 0.996i)9-s + (−0.221 + 0.0256i)10-s + (0.209 − 0.363i)11-s + (−0.997 − 0.0763i)12-s + 0.762·13-s + (0.304 − 0.952i)14-s + (0.0673 − 0.212i)15-s + (−0.0637 + 0.997i)16-s + (−1.22 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.534 - 0.844i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.534 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08583 + 0.597802i\)
\(L(\frac12)\) \(\approx\) \(1.08583 + 0.597802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.561i)T \)
3 \( 1 + (1.27 - 1.16i)T \)
7 \( 1 + (0.261 + 2.63i)T \)
good5 \( 1 + (0.432 - 0.249i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.695 + 1.20i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (5.04 + 2.91i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.14 - 1.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.53 + 4.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.32iT - 29T^{2} \)
31 \( 1 + (-5.98 - 3.45i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.67 + 6.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.74iT - 41T^{2} \)
43 \( 1 - 3.52iT - 43T^{2} \)
47 \( 1 + (-2.53 - 4.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.54 - 2.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.57 + 2.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.45 - 4.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.30 + 5.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.54T + 71T^{2} \)
73 \( 1 + (-4.98 + 8.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.84 + 1.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + (-2.12 + 1.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.526T + 97T^{2} \)
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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

−14.42277102586069767685273002952, −13.52020353930271227580821579015, −12.34219228820005812733581127288, −11.17757420039086540525515757763, −10.57131168799751132294792247462, −8.780339467099287732162246553368, −7.10321160765823228613685444763, −6.14456189761037719422754096065, −4.64858799110447758300727586014, −3.63063294412895302163054606797, 2.13605756298361284011251679922, 4.35925652640089098183772836868, 5.83235948350080755315369453418, 6.60861210262441450873925886296, 8.324735873989209646366027576015, 10.07676124749968817030281171541, 11.41069082486158563935731257528, 11.92002555992498771768224117454, 12.99933315499747855590973195584, 13.69033187964255442572687465889

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