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Properties

Label 2-84-28.27-c11-0-79
Degree $2$
Conductor $84$
Sign $-0.913 + 0.406i$
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  + (44.2 + 9.56i)2-s − 243·3-s + (1.86e3 + 845. i)4-s − 8.11e3i·5-s + (−1.07e4 − 2.32e3i)6-s + (−2.95e4 + 3.32e4i)7-s + (7.44e4 + 5.52e4i)8-s + 5.90e4·9-s + (7.76e4 − 3.59e5i)10-s − 8.13e4i·11-s + (−4.53e5 − 2.05e5i)12-s − 6.58e5i·13-s + (−1.62e6 + 1.18e6i)14-s + 1.97e6i·15-s + (2.76e6 + 3.15e6i)16-s − 1.82e5i·17-s + ⋯
L(s)  = 1  + (0.977 + 0.211i)2-s − 0.577·3-s + (0.910 + 0.413i)4-s − 1.16i·5-s + (−0.564 − 0.121i)6-s + (−0.664 + 0.747i)7-s + (0.802 + 0.596i)8-s + 0.333·9-s + (0.245 − 1.13i)10-s − 0.152i·11-s + (−0.525 − 0.238i)12-s − 0.491i·13-s + (−0.807 + 0.590i)14-s + 0.670i·15-s + (0.658 + 0.752i)16-s − 0.0312i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.913 + 0.406i$
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: \((2,\ 84,\ (\ :11/2),\ -0.913 + 0.406i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.131078 - 0.617524i\)
\(L(\frac12)\) \(\approx\) \(0.131078 - 0.617524i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-44.2 - 9.56i)T \)
3 \( 1 + 243T \)
7 \( 1 + (2.95e4 - 3.32e4i)T \)
good5 \( 1 + 8.11e3iT - 4.88e7T^{2} \)
11 \( 1 + 8.13e4iT - 2.85e11T^{2} \)
13 \( 1 + 6.58e5iT - 1.79e12T^{2} \)
17 \( 1 + 1.82e5iT - 3.42e13T^{2} \)
19 \( 1 + 2.23e6T + 1.16e14T^{2} \)
23 \( 1 + 2.20e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.52e8T + 1.22e16T^{2} \)
31 \( 1 + 7.39e7T + 2.54e16T^{2} \)
37 \( 1 + 5.56e8T + 1.77e17T^{2} \)
41 \( 1 - 7.10e8iT - 5.50e17T^{2} \)
43 \( 1 + 2.68e7iT - 9.29e17T^{2} \)
47 \( 1 + 1.89e9T + 2.47e18T^{2} \)
53 \( 1 + 2.21e9T + 9.26e18T^{2} \)
59 \( 1 + 7.11e9T + 3.01e19T^{2} \)
61 \( 1 + 5.76e9iT - 4.35e19T^{2} \)
67 \( 1 + 9.02e9iT - 1.22e20T^{2} \)
71 \( 1 + 2.87e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.24e10iT - 3.13e20T^{2} \)
79 \( 1 + 9.92e9iT - 7.47e20T^{2} \)
83 \( 1 + 2.76e10T + 1.28e21T^{2} \)
89 \( 1 + 7.34e9iT - 2.77e21T^{2} \)
97 \( 1 - 2.24e10iT - 7.15e21T^{2} \)
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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

−11.98714942526182247524237788710, −10.78018329731554872547193903722, −9.284270596636151306364553571208, −8.053303401729744487024174320584, −6.54293561976758905368172693744, −5.55939140402848337855415519636, −4.73988929544139494296885782957, −3.31000638483947028530527565292, −1.73379808499907966143334632211, −0.10708318564248999821095858252, 1.67743590574846934982070053546, 3.15623060168832060239231779789, 4.09732640259653447481747850563, 5.61077489364206081430521985137, 6.74271704666281556786681101217, 7.29002373329757154741669757123, 9.724529099009718020201250080689, 10.67926056330211270753169752676, 11.33830989400607164809711577966, 12.53602382728349124238440625535

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