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Properties

Label 2-84-12.11-c11-0-15
Degree $2$
Conductor $84$
Sign $-0.751 - 0.660i$
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  + (−33.5 + 30.3i)2-s + (342. + 244. i)3-s + (208. − 2.03e3i)4-s − 9.98e3i·5-s + (−1.89e4 + 2.19e3i)6-s − 1.68e4i·7-s + (5.47e4 + 7.47e4i)8-s + (5.78e4 + 1.67e5i)9-s + (3.02e5 + 3.35e5i)10-s − 6.42e5·11-s + (5.69e5 − 6.47e5i)12-s − 1.18e6·13-s + (5.09e5 + 5.64e5i)14-s + (2.43e6 − 3.42e6i)15-s + (−4.10e6 − 8.49e5i)16-s − 2.27e6i·17-s + ⋯
L(s)  = 1  + (−0.742 + 0.670i)2-s + (0.814 + 0.580i)3-s + (0.101 − 0.994i)4-s − 1.42i·5-s + (−0.993 + 0.115i)6-s − 0.377i·7-s + (0.591 + 0.806i)8-s + (0.326 + 0.945i)9-s + (0.957 + 1.06i)10-s − 1.20·11-s + (0.660 − 0.751i)12-s − 0.885·13-s + (0.253 + 0.280i)14-s + (0.829 − 1.16i)15-s + (−0.979 − 0.202i)16-s − 0.389i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.751 - 0.660i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.751 - 0.660i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.247843 + 0.657316i\)
\(L(\frac12)\) \(\approx\) \(0.247843 + 0.657316i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (33.5 - 30.3i)T \)
3 \( 1 + (-342. - 244. i)T \)
7 \( 1 + 1.68e4iT \)
good5 \( 1 + 9.98e3iT - 4.88e7T^{2} \)
11 \( 1 + 6.42e5T + 2.85e11T^{2} \)
13 \( 1 + 1.18e6T + 1.79e12T^{2} \)
17 \( 1 + 2.27e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.32e7iT - 1.16e14T^{2} \)
23 \( 1 + 1.06e7T + 9.52e14T^{2} \)
29 \( 1 - 1.12e8iT - 1.22e16T^{2} \)
31 \( 1 - 3.10e8iT - 2.54e16T^{2} \)
37 \( 1 - 5.44e8T + 1.77e17T^{2} \)
41 \( 1 + 4.69e8iT - 5.50e17T^{2} \)
43 \( 1 - 1.54e9iT - 9.29e17T^{2} \)
47 \( 1 - 3.01e8T + 2.47e18T^{2} \)
53 \( 1 - 1.24e9iT - 9.26e18T^{2} \)
59 \( 1 + 5.94e9T + 3.01e19T^{2} \)
61 \( 1 - 7.93e9T + 4.35e19T^{2} \)
67 \( 1 - 1.38e10iT - 1.22e20T^{2} \)
71 \( 1 - 1.99e9T + 2.31e20T^{2} \)
73 \( 1 - 1.98e10T + 3.13e20T^{2} \)
79 \( 1 - 7.26e9iT - 7.47e20T^{2} \)
83 \( 1 + 5.23e9T + 1.28e21T^{2} \)
89 \( 1 - 3.04e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.85e10T + 7.15e21T^{2} \)
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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

−12.70360221869434215759319702773, −10.90602514646633474598964247720, −9.848430725722765381628808814550, −9.024736832268300548731681257594, −8.176220308718164712032115942889, −7.22908865679407979507062025001, −5.16417252542948391078412278372, −4.69709575628856602590390876980, −2.58019049794543554867162884939, −1.08488764682819357337669037845, 0.20921118245963073478939990864, 2.13210774413664490369447825198, 2.60822337422359246124355037823, 3.79256693199799476193154173578, 6.23912658062475322306758643312, 7.58310465266598007873725044034, 8.008897811951981255313766135408, 9.623790763930784899660463175983, 10.30969484140786304673033343908, 11.53408804017387456268294922782

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