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Properties

Label 2-84-4.3-c2-0-6
Degree $2$
Conductor $84$
Sign $0.969 - 0.243i$
Analytic cond. $2.28883$
Root an. cond. $1.51288$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 − 0.245i)2-s + 1.73i·3-s + (3.87 − 0.973i)4-s + 0.424·5-s + (0.424 + 3.43i)6-s + 2.64i·7-s + (7.46 − 2.88i)8-s − 2.99·9-s + (0.842 − 0.104i)10-s − 6.71i·11-s + (1.68 + 6.71i)12-s − 9.57·13-s + (0.648 + 5.25i)14-s + 0.735i·15-s + (14.1 − 7.55i)16-s − 15.4·17-s + ⋯
L(s)  = 1  + (0.992 − 0.122i)2-s + 0.577i·3-s + (0.969 − 0.243i)4-s + 0.0848·5-s + (0.0707 + 0.572i)6-s + 0.377i·7-s + (0.932 − 0.360i)8-s − 0.333·9-s + (0.0842 − 0.0104i)10-s − 0.610i·11-s + (0.140 + 0.559i)12-s − 0.736·13-s + (0.0463 + 0.375i)14-s + 0.0490i·15-s + (0.881 − 0.472i)16-s − 0.909·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(2.28883\)
Root analytic conductor: \(1.51288\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.15953 + 0.266759i\)
\(L(\frac12)\) \(\approx\) \(2.15953 + 0.266759i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 + 0.245i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 - 2.64iT \)
good5 \( 1 - 0.424T + 25T^{2} \)
11 \( 1 + 6.71iT - 121T^{2} \)
13 \( 1 + 9.57T + 169T^{2} \)
17 \( 1 + 15.4T + 289T^{2} \)
19 \( 1 - 2.85iT - 361T^{2} \)
23 \( 1 + 32.6iT - 529T^{2} \)
29 \( 1 - 2.05T + 841T^{2} \)
31 \( 1 - 48.0iT - 961T^{2} \)
37 \( 1 - 18.5T + 1.36e3T^{2} \)
41 \( 1 - 24.5T + 1.68e3T^{2} \)
43 \( 1 - 81.2iT - 1.84e3T^{2} \)
47 \( 1 + 76.0iT - 2.20e3T^{2} \)
53 \( 1 - 64.9T + 2.80e3T^{2} \)
59 \( 1 - 46.0iT - 3.48e3T^{2} \)
61 \( 1 - 98.9T + 3.72e3T^{2} \)
67 \( 1 + 45.8iT - 4.48e3T^{2} \)
71 \( 1 - 114. iT - 5.04e3T^{2} \)
73 \( 1 - 11.0T + 5.32e3T^{2} \)
79 \( 1 + 7.63iT - 6.24e3T^{2} \)
83 \( 1 + 4.16iT - 6.88e3T^{2} \)
89 \( 1 + 159.T + 7.92e3T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−14.17509701716762616613791438614, −13.04026280781508557595192406238, −11.97171063286744949479272143027, −10.98401670027722613100141092884, −9.924251751201832287692926814777, −8.437612299609704708071579748730, −6.73959361301817004680280937981, −5.48279916317088710602416343438, −4.25254586052184821311267555626, −2.63163891135784931554856886600, 2.22761662924568131709346729992, 4.13292822942152883559333305447, 5.61179530072596554839298688898, 6.95535319977733166460021937833, 7.79336053725885750256895865308, 9.665085584624206283937309836551, 11.12592528998858223830292847340, 12.03497806792529807427612626310, 13.10401277046617354418610192430, 13.77293092674535210754716727277

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