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Properties

Label 2-84-21.20-c11-0-1
Degree $2$
Conductor $84$
Sign $-0.356 - 0.934i$
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  + (−402. − 123. i)3-s − 1.00e4·5-s + (2.73e4 + 3.50e4i)7-s + (1.46e5 + 9.93e4i)9-s + 2.46e5i·11-s − 1.47e6i·13-s + (4.04e6 + 1.24e6i)15-s − 2.04e6·17-s − 2.01e7i·19-s + (−6.66e6 − 1.74e7i)21-s − 3.93e7i·23-s + 5.22e7·25-s + (−4.67e7 − 5.80e7i)27-s + 1.09e8i·29-s + 1.28e8i·31-s + ⋯
L(s)  = 1  + (−0.955 − 0.293i)3-s − 1.43·5-s + (0.614 + 0.788i)7-s + (0.827 + 0.561i)9-s + 0.461i·11-s − 1.09i·13-s + (1.37 + 0.422i)15-s − 0.349·17-s − 1.86i·19-s + (−0.356 − 0.934i)21-s − 1.27i·23-s + 1.07·25-s + (−0.626 − 0.779i)27-s + 0.993i·29-s + 0.806i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.356 - 0.934i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.356 - 0.934i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.215094 + 0.312179i\)
\(L(\frac12)\) \(\approx\) \(0.215094 + 0.312179i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (402. + 123. i)T \)
7 \( 1 + (-2.73e4 - 3.50e4i)T \)
good5 \( 1 + 1.00e4T + 4.88e7T^{2} \)
11 \( 1 - 2.46e5iT - 2.85e11T^{2} \)
13 \( 1 + 1.47e6iT - 1.79e12T^{2} \)
17 \( 1 + 2.04e6T + 3.42e13T^{2} \)
19 \( 1 + 2.01e7iT - 1.16e14T^{2} \)
23 \( 1 + 3.93e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.09e8iT - 1.22e16T^{2} \)
31 \( 1 - 1.28e8iT - 2.54e16T^{2} \)
37 \( 1 + 1.17e8T + 1.77e17T^{2} \)
41 \( 1 - 1.13e9T + 5.50e17T^{2} \)
43 \( 1 - 1.02e9T + 9.29e17T^{2} \)
47 \( 1 + 1.19e9T + 2.47e18T^{2} \)
53 \( 1 + 2.82e9iT - 9.26e18T^{2} \)
59 \( 1 - 4.66e9T + 3.01e19T^{2} \)
61 \( 1 - 5.89e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.10e10T + 1.22e20T^{2} \)
71 \( 1 + 1.11e9iT - 2.31e20T^{2} \)
73 \( 1 + 6.64e9iT - 3.13e20T^{2} \)
79 \( 1 - 3.28e10T + 7.47e20T^{2} \)
83 \( 1 + 5.28e10T + 1.28e21T^{2} \)
89 \( 1 + 6.39e10T + 2.77e21T^{2} \)
97 \( 1 + 3.80e10iT - 7.15e21T^{2} \)
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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

−12.26867450643956812432416676303, −11.32383822005291012130918620103, −10.66522853719845611570012394759, −8.821620090351921861694262466806, −7.75026869974070801825854755691, −6.78340568612662115708796407751, −5.23370935677947302762111109403, −4.41013249682837645129892426318, −2.63357655896123450797688784093, −0.862818755046961206928241572388, 0.15382754698901954784601045965, 1.38403945905016331666702809351, 3.86900966767274928982181098656, 4.28358731275455016932548350317, 5.84829273241296899775455330695, 7.24109664669064999973672667391, 8.058629557643949844295768260324, 9.678814300941552001190860073755, 11.01783458477555399876834244697, 11.48795453260693373944669855996

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