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Properties

Label 2-84-84.23-c1-0-4
Degree $2$
Conductor $84$
Sign $0.866 - 0.499i$
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  + (−0.430 + 1.34i)2-s + (0.420 − 1.68i)3-s + (−1.62 − 1.16i)4-s + (2.36 + 1.36i)5-s + (2.08 + 1.29i)6-s + (1.89 + 1.84i)7-s + (2.26 − 1.69i)8-s + (−2.64 − 1.41i)9-s + (−2.85 + 2.59i)10-s + (−1.02 − 1.76i)11-s + (−2.63 + 2.24i)12-s − 4.44·13-s + (−3.30 + 1.76i)14-s + (3.28 − 3.39i)15-s + (1.30 + 3.78i)16-s + (−0.571 + 0.330i)17-s + ⋯
L(s)  = 1  + (−0.304 + 0.952i)2-s + (0.242 − 0.970i)3-s + (−0.814 − 0.580i)4-s + (1.05 + 0.609i)5-s + (0.850 + 0.526i)6-s + (0.717 + 0.696i)7-s + (0.800 − 0.598i)8-s + (−0.882 − 0.471i)9-s + (−0.902 + 0.820i)10-s + (−0.307 − 0.532i)11-s + (−0.760 + 0.649i)12-s − 1.23·13-s + (−0.882 + 0.470i)14-s + (0.848 − 0.876i)15-s + (0.326 + 0.945i)16-s + (−0.138 + 0.0800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913431 + 0.244215i\)
\(L(\frac12)\) \(\approx\) \(0.913431 + 0.244215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.430 - 1.34i)T \)
3 \( 1 + (-0.420 + 1.68i)T \)
7 \( 1 + (-1.89 - 1.84i)T \)
good5 \( 1 + (-2.36 - 1.36i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.02 + 1.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + (0.571 - 0.330i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.54 + 1.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.521 - 0.902i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.06iT - 29T^{2} \)
31 \( 1 + (2.68 - 1.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.76 + 8.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.85iT - 41T^{2} \)
43 \( 1 + 0.530iT - 43T^{2} \)
47 \( 1 + (-0.521 + 0.902i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.16 + 5.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.74 + 3.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0992 + 0.171i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.12 + 2.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.52T + 71T^{2} \)
73 \( 1 + (-1.51 - 2.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.45 - 5.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.15T + 83T^{2} \)
89 \( 1 + (0.468 + 0.270i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−14.48099583900874305092081692313, −13.61206182791936432518923945381, −12.57233176027369532295417086430, −11.01080016452674904127323029535, −9.611176961379418763447210028756, −8.545443654309715759313736848086, −7.42937355245095988623791890963, −6.29162267261951691269390948080, −5.30661007623381606665238821750, −2.25840379201142614671954231218, 2.23705198975076089072450034648, 4.30693300368748267148342537409, 5.21992656695010379511988152483, 7.78825137097523164928375555436, 9.030925840831452775166255539500, 9.940670334552564641866340932459, 10.56635060985983595716782466195, 11.89025562282635772289581594115, 13.14809762440908302732894251137, 14.03584722396631249910587798179

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