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Properties

Label 2-84-21.20-c11-0-3
Degree $2$
Conductor $84$
Sign $-0.994 + 0.103i$
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  + (300. + 294. i)3-s + 5.02e3·5-s + (−2.83e4 + 3.42e4i)7-s + (3.45e3 + 1.77e5i)9-s + 3.76e5i·11-s − 1.35e5i·13-s + (1.50e6 + 1.48e6i)15-s − 7.23e6·17-s − 5.82e6i·19-s + (−1.86e7 + 1.93e6i)21-s − 1.30e6i·23-s − 2.35e7·25-s + (−5.11e7 + 5.42e7i)27-s + 8.52e7i·29-s − 8.26e7i·31-s + ⋯
L(s)  = 1  + (0.713 + 0.700i)3-s + 0.718·5-s + (−0.637 + 0.770i)7-s + (0.0195 + 0.999i)9-s + 0.704i·11-s − 0.101i·13-s + (0.513 + 0.503i)15-s − 1.23·17-s − 0.539i·19-s + (−0.994 + 0.103i)21-s − 0.0421i·23-s − 0.483·25-s + (−0.686 + 0.727i)27-s + 0.772i·29-s − 0.518i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.994 + 0.103i$
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: \((2,\ 84,\ (\ :11/2),\ -0.994 + 0.103i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0724192 - 1.39569i\)
\(L(\frac12)\) \(\approx\) \(0.0724192 - 1.39569i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-300. - 294. i)T \)
7 \( 1 + (2.83e4 - 3.42e4i)T \)
good5 \( 1 - 5.02e3T + 4.88e7T^{2} \)
11 \( 1 - 3.76e5iT - 2.85e11T^{2} \)
13 \( 1 + 1.35e5iT - 1.79e12T^{2} \)
17 \( 1 + 7.23e6T + 3.42e13T^{2} \)
19 \( 1 + 5.82e6iT - 1.16e14T^{2} \)
23 \( 1 + 1.30e6iT - 9.52e14T^{2} \)
29 \( 1 - 8.52e7iT - 1.22e16T^{2} \)
31 \( 1 + 8.26e7iT - 2.54e16T^{2} \)
37 \( 1 - 5.72e8T + 1.77e17T^{2} \)
41 \( 1 + 2.33e8T + 5.50e17T^{2} \)
43 \( 1 + 1.16e9T + 9.29e17T^{2} \)
47 \( 1 - 2.56e8T + 2.47e18T^{2} \)
53 \( 1 + 1.13e9iT - 9.26e18T^{2} \)
59 \( 1 - 5.04e9T + 3.01e19T^{2} \)
61 \( 1 - 4.16e9iT - 4.35e19T^{2} \)
67 \( 1 + 5.89e9T + 1.22e20T^{2} \)
71 \( 1 + 1.31e10iT - 2.31e20T^{2} \)
73 \( 1 - 3.07e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.61e10T + 7.47e20T^{2} \)
83 \( 1 + 4.28e10T + 1.28e21T^{2} \)
89 \( 1 + 7.80e10T + 2.77e21T^{2} \)
97 \( 1 - 4.16e10iT - 7.15e21T^{2} \)
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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

−12.87423808559414378981250547668, −11.36922616239751029834984012341, −10.05268736792286411052293229733, −9.390252043989221701446418707371, −8.470730746028505904002280679743, −6.88904285700829648923599144763, −5.55336470987253433454699027119, −4.32575847273951276982413120194, −2.82952939634438007069753755663, −1.96888599815065834265122276952, 0.27737275206142892163861996640, 1.57279909984940457694051520362, 2.80471509582848147729576760712, 4.04688837633232085185817251770, 6.00049329566990411471333901313, 6.83738640831771615847411580108, 8.090710114147434264410980026972, 9.221611214225968670968706048393, 10.16833192286840333334549230672, 11.53895599838864988779683330844

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