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Properties

Label 2-84-7.5-c8-0-1
Degree $2$
Conductor $84$
Sign $0.00112 - 0.999i$
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  + (40.5 − 23.3i)3-s + (−939. − 542. i)5-s + (1.45e3 + 1.91e3i)7-s + (1.09e3 − 1.89e3i)9-s + (−8.43e3 − 1.46e4i)11-s − 9.95e3i·13-s − 5.07e4·15-s + (−9.17e4 + 5.29e4i)17-s + (1.47e5 + 8.52e4i)19-s + (1.03e5 + 4.35e4i)21-s + (−1.05e5 + 1.83e5i)23-s + (3.93e5 + 6.81e5i)25-s − 1.02e5i·27-s − 5.67e5·29-s + (5.49e5 − 3.17e5i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−1.50 − 0.867i)5-s + (0.604 + 0.796i)7-s + (0.166 − 0.288i)9-s + (−0.576 − 0.997i)11-s − 0.348i·13-s − 1.00·15-s + (−1.09 + 0.634i)17-s + (1.13 + 0.654i)19-s + (0.532 + 0.223i)21-s + (−0.377 + 0.654i)23-s + (1.00 + 1.74i)25-s − 0.192i·27-s − 0.802·29-s + (0.594 − 0.343i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00112 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.00112 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.00112 - 0.999i$
Analytic conductor: \(34.2198\)
Root analytic conductor: \(5.84976\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :4),\ 0.00112 - 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.563958 + 0.563325i\)
\(L(\frac12)\) \(\approx\) \(0.563958 + 0.563325i\)
\(L(5)\) 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 \( 1 + (-40.5 + 23.3i)T \)
7 \( 1 + (-1.45e3 - 1.91e3i)T \)
good5 \( 1 + (939. + 542. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (8.43e3 + 1.46e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 9.95e3iT - 8.15e8T^{2} \)
17 \( 1 + (9.17e4 - 5.29e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.47e5 - 8.52e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.05e5 - 1.83e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 5.67e5T + 5.00e11T^{2} \)
31 \( 1 + (-5.49e5 + 3.17e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (1.07e6 - 1.86e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 5.45e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.56e6T + 1.16e13T^{2} \)
47 \( 1 + (-2.76e6 - 1.59e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-4.38e6 - 7.59e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (8.08e6 - 4.66e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (2.13e7 + 1.23e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (6.91e6 + 1.19e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 1.19e7T + 6.45e14T^{2} \)
73 \( 1 + (1.83e7 - 1.05e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (8.61e6 - 1.49e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 6.82e7iT - 2.25e15T^{2} \)
89 \( 1 + (7.87e7 + 4.54e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 3.95e7iT - 7.83e15T^{2} \)
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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

−12.77833739572071154422881470472, −11.87557090663713940863497661024, −11.06539825112049520128166383589, −9.168088890542039145661322808188, −8.197817809003782810955212429903, −7.74961149695224943727624543543, −5.73773082818616937044581275238, −4.38376459624836875403019913029, −3.06409991206221839392531208834, −1.24410173329229203177388851349, 0.24268029182992125843475931388, 2.42358886350465002125477018834, 3.84806730822391613463631428532, 4.70520610767521455093426315833, 7.25621281005117916549213561795, 7.41565199389202168708294603167, 8.885960272995898647748527441070, 10.40180376301737596396329000264, 11.15146133594029177927670350216, 12.16985545005332987668637808127

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