Properties

Label 2-84-21.2-c8-0-14
Degree $2$
Conductor $84$
Sign $0.713 - 0.700i$
Analytic cond. $34.2198$
Root an. cond. $5.84976$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (51.3 + 62.6i)3-s + (918. + 530. i)5-s + (−1.25e3 − 2.04e3i)7-s + (−1.29e3 + 6.43e3i)9-s + (2.20e4 − 1.27e4i)11-s + 4.82e4·13-s + (1.39e4 + 8.48e4i)15-s + (1.51e4 − 8.72e3i)17-s + (6.76e4 − 1.17e5i)19-s + (6.37e4 − 1.83e5i)21-s + (−4.28e5 − 2.47e5i)23-s + (3.67e5 + 6.36e5i)25-s + (−4.69e5 + 2.49e5i)27-s + 4.48e5i·29-s + (1.92e5 + 3.32e5i)31-s + ⋯
L(s)  = 1  + (0.633 + 0.773i)3-s + (1.47 + 0.848i)5-s + (−0.523 − 0.852i)7-s + (−0.196 + 0.980i)9-s + (1.50 − 0.870i)11-s + 1.69·13-s + (0.275 + 1.67i)15-s + (0.180 − 0.104i)17-s + (0.519 − 0.899i)19-s + (0.327 − 0.944i)21-s + (−1.52 − 0.883i)23-s + (0.941 + 1.63i)25-s + (−0.883 + 0.469i)27-s + 0.633i·29-s + (0.207 + 0.360i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(34.2198\)
Root analytic conductor: \(5.84976\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :4),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.36282 + 1.37565i\)
\(L(\frac12)\) \(\approx\) \(3.36282 + 1.37565i\)
\(L(5)\) 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 \( 1 + (-51.3 - 62.6i)T \)
7 \( 1 + (1.25e3 + 2.04e3i)T \)
good5 \( 1 + (-918. - 530. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-2.20e4 + 1.27e4i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 4.82e4T + 8.15e8T^{2} \)
17 \( 1 + (-1.51e4 + 8.72e3i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-6.76e4 + 1.17e5i)T + (-8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (4.28e5 + 2.47e5i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 4.48e5iT - 5.00e11T^{2} \)
31 \( 1 + (-1.92e5 - 3.32e5i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (4.05e4 - 7.02e4i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 9.81e5iT - 7.98e12T^{2} \)
43 \( 1 - 1.23e6T + 1.16e13T^{2} \)
47 \( 1 + (2.72e6 + 1.57e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-6.10e5 + 3.52e5i)T + (3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (1.15e7 - 6.67e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-5.42e5 + 9.39e5i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-1.26e7 - 2.19e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 3.64e7iT - 6.45e14T^{2} \)
73 \( 1 + (4.68e6 + 8.10e6i)T + (-4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-8.25e5 + 1.42e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 3.97e7iT - 2.25e15T^{2} \)
89 \( 1 + (2.86e7 + 1.65e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 1.05e8T + 7.83e15T^{2} \)
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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

−13.35622155709671460148389155819, −11.23492574330973831383223316841, −10.42513349949007508027390977556, −9.543482006796155897954645200693, −8.616861302139691362342123405709, −6.73865861340088806781463452960, −5.91559733145455724550663959162, −3.95688585684239748108077323589, −2.99559594392777299468125818306, −1.34185978436340509538241738382, 1.31381404713263864337341740331, 1.94342752231198759826535644719, 3.74171056840942327924658131494, 5.86678531337013412022362032377, 6.34500774522334303943461605990, 8.201377158123173869305757557867, 9.271792696694358257298306862386, 9.696314689184258644119015952283, 11.88114407786603916971791524944, 12.59712979028593374491707721415

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