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Properties

Label 2-84-21.17-c1-0-2
Degree $2$
Conductor $84$
Sign $0.444 + 0.895i$
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 + (−1.5 − 2.59i)5-s + (2 + 1.73i)7-s − 2.99·9-s + (4.5 + 2.59i)11-s + (−4.5 + 2.59i)15-s + (−1.5 + 2.59i)17-s + (1.5 − 0.866i)19-s + (2.99 − 3.46i)21-s + (−4.5 + 2.59i)23-s + (−2 + 3.46i)25-s + 5.19i·27-s + (−1.5 − 0.866i)31-s + (4.5 − 7.79i)33-s + (1.5 − 7.79i)35-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.670 − 1.16i)5-s + (0.755 + 0.654i)7-s − 0.999·9-s + (1.35 + 0.783i)11-s + (−1.16 + 0.670i)15-s + (−0.363 + 0.630i)17-s + (0.344 − 0.198i)19-s + (0.654 − 0.755i)21-s + (−0.938 + 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.999i·27-s + (−0.269 − 0.155i)31-s + (0.783 − 1.35i)33-s + (0.253 − 1.31i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795524 - 0.493545i\)
\(L(\frac12)\) \(\approx\) \(0.795524 - 0.493545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.5 - 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 + 2.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−14.06615680336555087338634410833, −12.69589717085870242385032194163, −12.11139840696310783391186017406, −11.35856485317969817843575583503, −9.215859163200377806247973577386, −8.414372359329517044140273470889, −7.34333967932112187354646184389, −5.78976907741528653497454627437, −4.29558771480786242961523833261, −1.66083755492376087956416999134, 3.34339564043339402970541273188, 4.42674995746821934470783251129, 6.29060002996217269992566457595, 7.65784074094312859218918253664, 8.959569177307192509690606165204, 10.31977830992065478285507198937, 11.19547474441001663919566442074, 11.77099264527247962141301237020, 14.10908677261187547868823734370, 14.30007466802251090881176146207

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