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Properties

Label 2-84-28.19-c1-0-2
Degree $2$
Conductor $84$
Sign $0.934 - 0.355i$
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  + (−1.06 + 0.929i)2-s + (−0.5 − 0.866i)3-s + (0.272 − 1.98i)4-s + (2.12 + 1.22i)5-s + (1.33 + 0.458i)6-s + (2.63 − 0.272i)7-s + (1.55 + 2.36i)8-s + (−0.499 + 0.866i)9-s + (−3.40 + 0.667i)10-s + (−1.09 + 0.632i)11-s + (−1.85 + 0.755i)12-s − 2.99i·13-s + (−2.55 + 2.73i)14-s − 2.45i·15-s + (−3.85 − 1.07i)16-s + (1.58 − 0.916i)17-s + ⋯
L(s)  = 1  + (−0.753 + 0.657i)2-s + (−0.288 − 0.499i)3-s + (0.136 − 0.990i)4-s + (0.949 + 0.548i)5-s + (0.546 + 0.187i)6-s + (0.994 − 0.102i)7-s + (0.548 + 0.836i)8-s + (−0.166 + 0.288i)9-s + (−1.07 + 0.210i)10-s + (−0.330 + 0.190i)11-s + (−0.534 + 0.217i)12-s − 0.831i·13-s + (−0.681 + 0.731i)14-s − 0.633i·15-s + (−0.962 − 0.269i)16-s + (0.385 − 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.741565 + 0.136301i\)
\(L(\frac12)\) \(\approx\) \(0.741565 + 0.136301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 - 0.929i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.63 + 0.272i)T \)
good5 \( 1 + (-2.12 - 1.22i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.09 - 0.632i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
17 \( 1 + (-1.58 + 0.916i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.07 - 3.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.83 + 3.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + (-4.71 - 8.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.75 - 6.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.08iT - 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + (-3.67 + 6.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0358 - 0.0620i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.61 + 5.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.43 - 1.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (-7.01 + 4.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.54 + 0.891i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + (-7.42 - 4.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.10iT - 97T^{2} \)
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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

−14.37133723392280850695953913415, −13.66163726060888360185303595370, −12.09172135662330490698029830484, −10.64666049914349106989150970615, −10.12339305815492990800418656565, −8.463192410530064124968427841135, −7.52508972004096420438500648842, −6.23380891402184852396262249047, −5.24262750986828721801133371207, −1.93607887109344385434486206661, 1.95286796579453085706415655806, 4.27771932115628961675179378893, 5.77907436825075065321263815603, 7.70526731647213206810551586176, 8.982865482728242863606220487675, 9.727161114370042506434975681116, 10.94518121477555447540450773736, 11.72426193257430648840971214585, 13.00440718921444370887418115822, 14.03160466395604093035344631332

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