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Properties

Label 2-84-28.27-c11-0-50
Degree $2$
Conductor $84$
Sign $0.515 + 0.857i$
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  + (−3.91 + 45.0i)2-s − 243·3-s + (−2.01e3 − 352. i)4-s − 7.26e3i·5-s + (950. − 1.09e4i)6-s + (−2.91e4 − 3.35e4i)7-s + (2.37e4 − 8.95e4i)8-s + 5.90e4·9-s + (3.27e5 + 2.83e4i)10-s + 8.15e5i·11-s + (4.90e5 + 8.56e4i)12-s + 1.71e6i·13-s + (1.62e6 − 1.18e6i)14-s + 1.76e6i·15-s + (3.94e6 + 1.42e6i)16-s + 6.55e6i·17-s + ⋯
L(s)  = 1  + (−0.0864 + 0.996i)2-s − 0.577·3-s + (−0.985 − 0.172i)4-s − 1.03i·5-s + (0.0498 − 0.575i)6-s + (−0.655 − 0.755i)7-s + (0.256 − 0.966i)8-s + 0.333·9-s + (1.03 + 0.0898i)10-s + 1.52i·11-s + (0.568 + 0.0994i)12-s + 1.28i·13-s + (0.809 − 0.587i)14-s + 0.600i·15-s + (0.940 + 0.339i)16-s + 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.515 + 0.857i$
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: \((2,\ 84,\ (\ :11/2),\ 0.515 + 0.857i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.521785 - 0.295143i\)
\(L(\frac12)\) \(\approx\) \(0.521785 - 0.295143i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.91 - 45.0i)T \)
3 \( 1 + 243T \)
7 \( 1 + (2.91e4 + 3.35e4i)T \)
good5 \( 1 + 7.26e3iT - 4.88e7T^{2} \)
11 \( 1 - 8.15e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.71e6iT - 1.79e12T^{2} \)
17 \( 1 - 6.55e6iT - 3.42e13T^{2} \)
19 \( 1 + 6.01e5T + 1.16e14T^{2} \)
23 \( 1 + 4.85e7iT - 9.52e14T^{2} \)
29 \( 1 - 6.73e7T + 1.22e16T^{2} \)
31 \( 1 - 1.08e8T + 2.54e16T^{2} \)
37 \( 1 - 6.93e7T + 1.77e17T^{2} \)
41 \( 1 - 4.75e8iT - 5.50e17T^{2} \)
43 \( 1 + 1.48e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.19e9T + 2.47e18T^{2} \)
53 \( 1 - 5.89e8T + 9.26e18T^{2} \)
59 \( 1 + 1.03e10T + 3.01e19T^{2} \)
61 \( 1 + 5.20e9iT - 4.35e19T^{2} \)
67 \( 1 - 4.38e9iT - 1.22e20T^{2} \)
71 \( 1 - 1.15e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.02e10iT - 3.13e20T^{2} \)
79 \( 1 - 7.42e9iT - 7.47e20T^{2} \)
83 \( 1 + 5.40e9T + 1.28e21T^{2} \)
89 \( 1 + 6.36e9iT - 2.77e21T^{2} \)
97 \( 1 + 9.22e10iT - 7.15e21T^{2} \)
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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

−12.32548425336966559356340474356, −10.40508605500500788904680016282, −9.541793448449778502316361228130, −8.427350292156790606662591693085, −7.04580405826997738753722817945, −6.31336469370595163069104322736, −4.69502785464682425393539430239, −4.25293192444874979127871267723, −1.48360812453194873163268082942, −0.22594884173582213850187378872, 0.890341382399222700124520266120, 2.81540577735934904126847335924, 3.29131589567022677115164074236, 5.25968008474467866383960231836, 6.22128088829362219848705992015, 7.86232816651026969632724888552, 9.232695848546688296512857765642, 10.28791534355210880231087786113, 11.16044373254525729580254201540, 11.90028840767484160717659267183

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