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Properties

Label 2-84-21.20-c1-0-0
Degree $2$
Conductor $84$
Sign $0.654 - 0.755i$
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  + 1.73i·3-s + (2 + 1.73i)7-s − 2.99·9-s − 6.92i·13-s − 3.46i·19-s + (−2.99 + 3.46i)21-s − 5·25-s − 5.19i·27-s + 10.3i·31-s + 10·37-s + 11.9·39-s − 8·43-s + (1.00 + 6.92i)49-s + 5.99·57-s + 6.92i·61-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.755 + 0.654i)7-s − 0.999·9-s − 1.92i·13-s − 0.794i·19-s + (−0.654 + 0.755i)21-s − 25-s − 0.999i·27-s + 1.86i·31-s + 1.64·37-s + 1.92·39-s − 1.21·43-s + (0.142 + 0.989i)49-s + 0.794·57-s + 0.887i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893674 + 0.408275i\)
\(L(\frac12)\) \(\approx\) \(0.893674 + 0.408275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−14.80511695638160694849422515926, −13.45709188274793648937010381999, −12.11078666738757929154452669387, −11.03872821307273511904902517339, −10.13000869513673846219308250556, −8.860431203632680285379101442976, −7.889190764798384171636941586833, −5.80122190357005510453887893234, −4.81237983127692909102423173270, −3.01248503846707493834137147349, 1.84955169590426068716106906195, 4.24043619370547132373815289388, 6.07005500345546038890961276394, 7.27688550516460778984917889313, 8.231499877023010211585730486077, 9.626848095956417877996067048542, 11.29182449378243979210408310932, 11.81863185793182487827709098494, 13.24271376041736625582065078385, 14.01514784981660746800938046652

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