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Properties

Label 2-84-28.27-c11-0-16
Degree $2$
Conductor $84$
Sign $-0.798 + 0.601i$
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  + (33.6 + 30.2i)2-s − 243·3-s + (218. + 2.03e3i)4-s + 2.34e3i·5-s + (−8.18e3 − 7.34e3i)6-s + (2.28e4 + 3.81e4i)7-s + (−5.42e4 + 7.51e4i)8-s + 5.90e4·9-s + (−7.09e4 + 7.90e4i)10-s − 8.78e5i·11-s + (−5.31e4 − 4.94e5i)12-s + 2.43e6i·13-s + (−3.87e5 + 1.97e6i)14-s − 5.70e5i·15-s + (−4.09e6 + 8.90e5i)16-s + 7.78e6i·17-s + ⋯
L(s)  = 1  + (0.743 + 0.668i)2-s − 0.577·3-s + (0.106 + 0.994i)4-s + 0.335i·5-s + (−0.429 − 0.385i)6-s + (0.512 + 0.858i)7-s + (−0.585 + 0.810i)8-s + 0.333·9-s + (−0.224 + 0.249i)10-s − 1.64i·11-s + (−0.0616 − 0.574i)12-s + 1.82i·13-s + (−0.192 + 0.981i)14-s − 0.193i·15-s + (−0.977 + 0.212i)16-s + 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.798 + 0.601i$
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: \((2,\ 84,\ (\ :11/2),\ -0.798 + 0.601i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.542309 - 1.62196i\)
\(L(\frac12)\) \(\approx\) \(0.542309 - 1.62196i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-33.6 - 30.2i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-2.28e4 - 3.81e4i)T \)
good5 \( 1 - 2.34e3iT - 4.88e7T^{2} \)
11 \( 1 + 8.78e5iT - 2.85e11T^{2} \)
13 \( 1 - 2.43e6iT - 1.79e12T^{2} \)
17 \( 1 - 7.78e6iT - 3.42e13T^{2} \)
19 \( 1 + 5.19e6T + 1.16e14T^{2} \)
23 \( 1 - 2.60e7iT - 9.52e14T^{2} \)
29 \( 1 + 2.93e7T + 1.22e16T^{2} \)
31 \( 1 - 1.06e8T + 2.54e16T^{2} \)
37 \( 1 - 3.32e8T + 1.77e17T^{2} \)
41 \( 1 + 4.43e8iT - 5.50e17T^{2} \)
43 \( 1 + 9.39e8iT - 9.29e17T^{2} \)
47 \( 1 + 3.08e9T + 2.47e18T^{2} \)
53 \( 1 + 5.68e9T + 9.26e18T^{2} \)
59 \( 1 + 3.41e9T + 3.01e19T^{2} \)
61 \( 1 + 3.61e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.64e10iT - 1.22e20T^{2} \)
71 \( 1 - 9.96e9iT - 2.31e20T^{2} \)
73 \( 1 - 1.66e10iT - 3.13e20T^{2} \)
79 \( 1 + 1.88e10iT - 7.47e20T^{2} \)
83 \( 1 - 8.20e9T + 1.28e21T^{2} \)
89 \( 1 + 5.00e10iT - 2.77e21T^{2} \)
97 \( 1 - 7.38e8iT - 7.15e21T^{2} \)
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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

−12.70221171879919384991759807508, −11.57699395849882136537762287540, −11.06540081011096804932751109077, −9.033919770393471091347043459569, −8.121997425503577353930812761359, −6.55073515903501914140768329990, −5.94060686657596766328445603428, −4.69171292536197674723020185195, −3.41370413273938973873478805258, −1.83370276991294248587791581217, 0.35059843634858399574263484464, 1.33362383023603026069907268651, 2.84938178323366121089403167195, 4.56292988671581803574319050444, 4.97937384411649791119963177988, 6.56204794342993041525337720057, 7.77206634519825338239347464461, 9.710045538052702167060966216740, 10.45259134813000296394814395000, 11.42854007980322539175714658428

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