Properties

Label 2-84-7.5-c8-0-4
Degree $2$
Conductor $84$
Sign $0.962 + 0.272i$
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 + (−133. − 77.3i)5-s + (−2.23e3 + 877. i)7-s + (1.09e3 − 1.89e3i)9-s + (3.95e3 + 6.84e3i)11-s + 4.17e3i·13-s − 7.23e3·15-s + (1.10e5 − 6.38e4i)17-s + (1.39e5 + 8.03e4i)19-s + (−6.99e4 + 8.77e4i)21-s + (−1.64e4 + 2.84e4i)23-s + (−1.83e5 − 3.17e5i)25-s − 1.02e5i·27-s + 6.59e5·29-s + (−5.16e3 + 2.98e3i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.214 − 0.123i)5-s + (−0.930 + 0.365i)7-s + (0.166 − 0.288i)9-s + (0.269 + 0.467i)11-s + 0.146i·13-s − 0.142·15-s + (1.32 − 0.764i)17-s + (1.06 + 0.616i)19-s + (−0.359 + 0.451i)21-s + (−0.0586 + 0.101i)23-s + (−0.469 − 0.813i)25-s − 0.192i·27-s + 0.932·29-s + (−0.00559 + 0.00322i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.962 + 0.272i$
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: \((2,\ 84,\ (\ :4),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.15898 - 0.300046i\)
\(L(\frac12)\) \(\approx\) \(2.15898 - 0.300046i\)
\(L(5)\) 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 \( 1 + (2.23e3 - 877. i)T \)
good5 \( 1 + (133. + 77.3i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-3.95e3 - 6.84e3i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 4.17e3iT - 8.15e8T^{2} \)
17 \( 1 + (-1.10e5 + 6.38e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.39e5 - 8.03e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.64e4 - 2.84e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 6.59e5T + 5.00e11T^{2} \)
31 \( 1 + (5.16e3 - 2.98e3i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-3.29e5 + 5.70e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 1.10e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.57e6T + 1.16e13T^{2} \)
47 \( 1 + (1.35e6 + 7.84e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-4.31e6 - 7.46e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.85e7 + 1.07e7i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-2.16e7 - 1.25e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (9.65e6 + 1.67e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 9.50e6T + 6.45e14T^{2} \)
73 \( 1 + (-6.91e6 + 3.99e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (1.32e6 - 2.29e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 5.37e7iT - 2.25e15T^{2} \)
89 \( 1 + (3.33e7 + 1.92e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 2.56e7iT - 7.83e15T^{2} \)
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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

−12.42925612881154766287663640769, −11.90688520018817657055566099650, −10.06547771610212932681563025021, −9.339391253085444849136714792030, −8.004788411786501465919141789902, −6.93218427305231370759596620206, −5.57205010079561531293601325292, −3.81854939964350195386387680580, −2.60164516289879544294932819825, −0.878128059882366444452079891846, 0.932732940275572432934512714706, 2.96483844625431710978486573100, 3.86780347647707147798574219525, 5.61095334493161863459983383233, 7.01923051779239429535378208720, 8.177876101233350547635231081585, 9.469899127529163033284928887510, 10.27850410993525063303315832119, 11.58066779527270606031131202043, 12.81320082345020143943884132394

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