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Properties

Label 2-84-28.27-c11-0-58
Degree $2$
Conductor $84$
Sign $0.374 + 0.927i$
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  + (−39.5 − 21.9i)2-s − 243·3-s + (1.08e3 + 1.73e3i)4-s + 1.02e4i·5-s + (9.61e3 + 5.33e3i)6-s + (4.37e4 + 7.75e3i)7-s + (−4.85e3 − 9.25e4i)8-s + 5.90e4·9-s + (2.25e5 − 4.07e5i)10-s + 2.63e5i·11-s + (−2.63e5 − 4.22e5i)12-s − 3.18e5i·13-s + (−1.56e6 − 1.26e6i)14-s − 2.50e6i·15-s + (−1.83e6 + 3.77e6i)16-s − 8.54e6i·17-s + ⋯
L(s)  = 1  + (−0.874 − 0.484i)2-s − 0.577·3-s + (0.529 + 0.848i)4-s + 1.47i·5-s + (0.504 + 0.279i)6-s + (0.984 + 0.174i)7-s + (−0.0524 − 0.998i)8-s + 0.333·9-s + (0.713 − 1.28i)10-s + 0.492i·11-s + (−0.305 − 0.489i)12-s − 0.237i·13-s + (−0.776 − 0.629i)14-s − 0.850i·15-s + (−0.438 + 0.898i)16-s − 1.45i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.374 + 0.927i$
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: \((2,\ 84,\ (\ :11/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.673972 - 0.454914i\)
\(L(\frac12)\) \(\approx\) \(0.673972 - 0.454914i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (39.5 + 21.9i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-4.37e4 - 7.75e3i)T \)
good5 \( 1 - 1.02e4iT - 4.88e7T^{2} \)
11 \( 1 - 2.63e5iT - 2.85e11T^{2} \)
13 \( 1 + 3.18e5iT - 1.79e12T^{2} \)
17 \( 1 + 8.54e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.17e7T + 1.16e14T^{2} \)
23 \( 1 + 2.52e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.70e8T + 1.22e16T^{2} \)
31 \( 1 - 7.50e7T + 2.54e16T^{2} \)
37 \( 1 + 3.97e8T + 1.77e17T^{2} \)
41 \( 1 + 1.19e9iT - 5.50e17T^{2} \)
43 \( 1 - 5.04e8iT - 9.29e17T^{2} \)
47 \( 1 + 3.10e9T + 2.47e18T^{2} \)
53 \( 1 + 1.83e9T + 9.26e18T^{2} \)
59 \( 1 - 5.07e9T + 3.01e19T^{2} \)
61 \( 1 - 1.56e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.34e10iT - 1.22e20T^{2} \)
71 \( 1 + 1.82e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.33e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.25e10iT - 7.47e20T^{2} \)
83 \( 1 + 4.17e10T + 1.28e21T^{2} \)
89 \( 1 + 6.38e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.84e10iT - 7.15e21T^{2} \)
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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

−11.49478992142594016987953141710, −10.72655338518621337161658370143, −10.00627284754833833250571731355, −8.473424956670930250258369128403, −7.27670132421205964320183629039, −6.49582926751800137510329056359, −4.63953567671694351397294928563, −2.95703291565034961001497161047, −1.94438515428506577670819650573, −0.34562127864135076022730721396, 0.988039357737073313052840532979, 1.68780106907574870090939017318, 4.43394044136690153821862526323, 5.37061340185213442352693853936, 6.51141023587035794621566647447, 8.216541065600073296231608401328, 8.513415794679411161429879374132, 9.964263953298172762553811298386, 11.04331360585985929281964773608, 11.99997858803416418854295039270

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