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Properties

Label 2-84-21.20-c11-0-8
Degree $2$
Conductor $84$
Sign $0.503 + 0.864i$
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  + (24.0 − 420. i)3-s − 1.19e4·5-s + (−3.70e4 + 2.45e4i)7-s + (−1.75e5 − 2.02e4i)9-s + 2.58e5i·11-s + 1.86e6i·13-s + (−2.86e5 + 5.01e6i)15-s − 8.63e6·17-s − 1.34e7i·19-s + (9.41e6 + 1.61e7i)21-s + 3.97e7i·23-s + 9.33e7·25-s + (−1.27e7 + 7.34e7i)27-s − 1.49e8i·29-s + 4.61e7i·31-s + ⋯
L(s)  = 1  + (0.0571 − 0.998i)3-s − 1.70·5-s + (−0.833 + 0.551i)7-s + (−0.993 − 0.114i)9-s + 0.484i·11-s + 1.39i·13-s + (−0.0974 + 1.70i)15-s − 1.47·17-s − 1.24i·19-s + (0.503 + 0.864i)21-s + 1.28i·23-s + 1.91·25-s + (−0.170 + 0.985i)27-s − 1.35i·29-s + 0.289i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.503 + 0.864i$
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: \((2,\ 84,\ (\ :11/2),\ 0.503 + 0.864i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.319998 - 0.183948i\)
\(L(\frac12)\) \(\approx\) \(0.319998 - 0.183948i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-24.0 + 420. i)T \)
7 \( 1 + (3.70e4 - 2.45e4i)T \)
good5 \( 1 + 1.19e4T + 4.88e7T^{2} \)
11 \( 1 - 2.58e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.86e6iT - 1.79e12T^{2} \)
17 \( 1 + 8.63e6T + 3.42e13T^{2} \)
19 \( 1 + 1.34e7iT - 1.16e14T^{2} \)
23 \( 1 - 3.97e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.49e8iT - 1.22e16T^{2} \)
31 \( 1 - 4.61e7iT - 2.54e16T^{2} \)
37 \( 1 + 9.06e7T + 1.77e17T^{2} \)
41 \( 1 + 1.17e9T + 5.50e17T^{2} \)
43 \( 1 + 3.46e8T + 9.29e17T^{2} \)
47 \( 1 + 2.49e9T + 2.47e18T^{2} \)
53 \( 1 - 1.36e9iT - 9.26e18T^{2} \)
59 \( 1 + 3.00e9T + 3.01e19T^{2} \)
61 \( 1 + 6.74e9iT - 4.35e19T^{2} \)
67 \( 1 + 2.90e8T + 1.22e20T^{2} \)
71 \( 1 - 1.33e10iT - 2.31e20T^{2} \)
73 \( 1 - 3.40e9iT - 3.13e20T^{2} \)
79 \( 1 - 3.51e10T + 7.47e20T^{2} \)
83 \( 1 + 5.87e9T + 1.28e21T^{2} \)
89 \( 1 + 9.42e8T + 2.77e21T^{2} \)
97 \( 1 + 1.56e11iT - 7.15e21T^{2} \)
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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

−11.66087408460647246068109858665, −11.45537508934016569919832067055, −9.306054410745308442378866097149, −8.398732260659069427790748646268, −7.15329119898228093054395705614, −6.57214902272562845264639298162, −4.64226173135875308044278311608, −3.34017891726128524756888846224, −2.01158424446413079259339278044, −0.23586866128152138600281946712, 0.39424803260383136614708726731, 3.13286654683944573541447111756, 3.74877429729349737085709614105, 4.87441064513714099209630100802, 6.52736697822545770157231929756, 7.961343107312453323143927467370, 8.738118678556334144664571177982, 10.30993326177176037078729918861, 10.89704887648814518967194567758, 12.07119636871793617517887728404

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