Properties

Label 2-84-7.5-c8-0-3
Degree $2$
Conductor $84$
Sign $0.225 - 0.974i$
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 + (632. + 365. i)5-s + (984. + 2.18e3i)7-s + (1.09e3 − 1.89e3i)9-s + (6.35e3 + 1.10e4i)11-s + 2.11e4i·13-s + 3.41e4·15-s + (−8.25e4 + 4.76e4i)17-s + (−1.33e5 − 7.69e4i)19-s + (9.10e4 + 6.56e4i)21-s + (−4.71e4 + 8.15e4i)23-s + (7.14e4 + 1.23e5i)25-s − 1.02e5i·27-s + 5.41e5·29-s + (−1.85e5 + 1.07e5i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (1.01 + 0.584i)5-s + (0.410 + 0.912i)7-s + (0.166 − 0.288i)9-s + (0.433 + 0.751i)11-s + 0.741i·13-s + 0.674·15-s + (−0.988 + 0.570i)17-s + (−1.02 − 0.590i)19-s + (0.468 + 0.337i)21-s + (−0.168 + 0.291i)23-s + (0.182 + 0.316i)25-s − 0.192i·27-s + 0.765·29-s + (−0.201 + 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.225 - 0.974i$
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: \((2,\ 84,\ (\ :4),\ 0.225 - 0.974i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.16422 + 1.71975i\)
\(L(\frac12)\) \(\approx\) \(2.16422 + 1.71975i\)
\(L(5)\) 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 \( 1 + (-984. - 2.18e3i)T \)
good5 \( 1 + (-632. - 365. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-6.35e3 - 1.10e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 2.11e4iT - 8.15e8T^{2} \)
17 \( 1 + (8.25e4 - 4.76e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.33e5 + 7.69e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (4.71e4 - 8.15e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 5.41e5T + 5.00e11T^{2} \)
31 \( 1 + (1.85e5 - 1.07e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (3.70e5 - 6.41e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 7.82e5iT - 7.98e12T^{2} \)
43 \( 1 + 2.21e5T + 1.16e13T^{2} \)
47 \( 1 + (-7.13e6 - 4.11e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-6.59e5 - 1.14e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (7.52e6 - 4.34e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.31e7 - 7.57e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.99e7 - 3.44e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 9.34e6T + 6.45e14T^{2} \)
73 \( 1 + (-2.94e7 + 1.70e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-2.26e7 + 3.92e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 6.58e7iT - 2.25e15T^{2} \)
89 \( 1 + (1.09e7 + 6.34e6i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 6.15e6iT - 7.83e15T^{2} \)
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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

−12.96715853477216538397288296419, −11.87302316422434469940346295753, −10.60049067273313692160854315428, −9.374940809925590389540347509284, −8.587335151113134437735436021386, −6.95067881729296740608667443335, −6.06741407199783422745234723761, −4.41907988757164802498274735896, −2.47245478673347349570379229669, −1.79697508301208897305381915796, 0.75434588099818830649234733394, 2.15759967470576661804300021652, 3.86567961113634071092691135845, 5.14166988964085291077443936527, 6.52310573182968975494889717313, 8.092507215434615528732534910382, 9.032589466229227319450692548917, 10.15479631191783430705214276289, 11.05430352076333646901557109937, 12.68981140912274944075768501892

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