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Properties

Label 2-84-28.27-c11-0-36
Degree $2$
Conductor $84$
Sign $-0.944 + 0.329i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (41.8 + 17.1i)2-s − 243·3-s + (1.45e3 + 1.43e3i)4-s + 1.03e4i·5-s + (−1.01e4 − 4.17e3i)6-s + (−1.95e4 + 3.99e4i)7-s + (3.62e4 + 8.53e4i)8-s + 5.90e4·9-s + (−1.78e5 + 4.33e5i)10-s + 9.48e5i·11-s + (−3.54e5 − 3.49e5i)12-s + 4.11e5i·13-s + (−1.50e6 + 1.33e6i)14-s − 2.51e6i·15-s + (5.06e4 + 4.19e6i)16-s − 1.70e6i·17-s + ⋯
L(s)  = 1  + (0.925 + 0.379i)2-s − 0.577·3-s + (0.711 + 0.702i)4-s + 1.48i·5-s + (−0.534 − 0.219i)6-s + (−0.440 + 0.897i)7-s + (0.391 + 0.920i)8-s + 0.333·9-s + (−0.563 + 1.37i)10-s + 1.77i·11-s + (−0.410 − 0.405i)12-s + 0.307i·13-s + (−0.748 + 0.663i)14-s − 0.856i·15-s + (0.0120 + 0.999i)16-s − 0.291i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.944 + 0.329i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.502692 - 2.97052i\)
\(L(\frac12)\) \(\approx\) \(0.502692 - 2.97052i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-41.8 - 17.1i)T \)
3 \( 1 + 243T \)
7 \( 1 + (1.95e4 - 3.99e4i)T \)
good5 \( 1 - 1.03e4iT - 4.88e7T^{2} \)
11 \( 1 - 9.48e5iT - 2.85e11T^{2} \)
13 \( 1 - 4.11e5iT - 1.79e12T^{2} \)
17 \( 1 + 1.70e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.17e7T + 1.16e14T^{2} \)
23 \( 1 - 4.13e6iT - 9.52e14T^{2} \)
29 \( 1 - 1.88e8T + 1.22e16T^{2} \)
31 \( 1 + 1.51e8T + 2.54e16T^{2} \)
37 \( 1 - 6.81e8T + 1.77e17T^{2} \)
41 \( 1 + 8.50e8iT - 5.50e17T^{2} \)
43 \( 1 - 1.82e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.56e9T + 2.47e18T^{2} \)
53 \( 1 + 1.37e9T + 9.26e18T^{2} \)
59 \( 1 + 5.42e9T + 3.01e19T^{2} \)
61 \( 1 - 8.36e9iT - 4.35e19T^{2} \)
67 \( 1 - 4.94e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.73e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.99e10iT - 3.13e20T^{2} \)
79 \( 1 + 1.02e10iT - 7.47e20T^{2} \)
83 \( 1 - 5.29e10T + 1.28e21T^{2} \)
89 \( 1 - 8.39e10iT - 2.77e21T^{2} \)
97 \( 1 + 6.83e10iT - 7.15e21T^{2} \)
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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

−12.42022662227932181389650569810, −11.84927887605993118570103127852, −10.70334085406455556178069109374, −9.553234027967577111920329347620, −7.48357879802960884405206505934, −6.81252861109160013626505387405, −5.82627135149747122341684081520, −4.55879221576959431114172278701, −3.07900184215445540093329639080, −2.11903046281526910145443746665, 0.68207489069887558740317899686, 1.05460571214358940562420540020, 3.21012737789209096598777054967, 4.37736463656660390774402287375, 5.41643389332820304426661901343, 6.34032017511294233516774070154, 7.961595587931932115465101911952, 9.443850621103679321721182702974, 10.65239261511953457804180444662, 11.58274645015460035380638749885

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