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Properties

Label 2-84-7.3-c8-0-4
Degree $2$
Conductor $84$
Sign $0.214 - 0.976i$
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 + (1.01e3 − 584. i)5-s + (−1.82e3 + 1.55e3i)7-s + (1.09e3 + 1.89e3i)9-s + (−1.30e4 + 2.26e4i)11-s + 2.85e4i·13-s + 5.46e4·15-s + (−1.80e4 − 1.04e4i)17-s + (4.29e4 − 2.48e4i)19-s + (−1.10e5 + 2.01e4i)21-s + (1.70e5 + 2.96e5i)23-s + (4.87e5 − 8.43e5i)25-s + 1.02e5i·27-s − 1.12e6·29-s + (1.47e6 + 8.51e5i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (1.61 − 0.934i)5-s + (−0.762 + 0.647i)7-s + (0.166 + 0.288i)9-s + (−0.893 + 1.54i)11-s + 1.00i·13-s + 1.07·15-s + (−0.215 − 0.124i)17-s + (0.329 − 0.190i)19-s + (−0.567 + 0.103i)21-s + (0.610 + 1.05i)23-s + (1.24 − 2.16i)25-s + 0.192i·27-s − 1.59·29-s + (1.59 + 0.922i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.214 - 0.976i$
Analytic conductor: \(34.2198\)
Root analytic conductor: \(5.84976\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :4),\ 0.214 - 0.976i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.99820 + 1.60702i\)
\(L(\frac12)\) \(\approx\) \(1.99820 + 1.60702i\)
\(L(5)\) 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 \( 1 + (1.82e3 - 1.55e3i)T \)
good5 \( 1 + (-1.01e3 + 584. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (1.30e4 - 2.26e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 2.85e4iT - 8.15e8T^{2} \)
17 \( 1 + (1.80e4 + 1.04e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-4.29e4 + 2.48e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-1.70e5 - 2.96e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 1.12e6T + 5.00e11T^{2} \)
31 \( 1 + (-1.47e6 - 8.51e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-3.20e5 - 5.54e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 1.34e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.80e6T + 1.16e13T^{2} \)
47 \( 1 + (-7.88e6 + 4.55e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (1.82e6 - 3.16e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (-9.97e6 - 5.76e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (8.77e6 - 5.06e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (8.86e6 - 1.53e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 3.79e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.94e7 - 1.70e7i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (9.66e6 + 1.67e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 9.60e5iT - 2.25e15T^{2} \)
89 \( 1 + (-5.62e7 + 3.24e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 2.66e7iT - 7.83e15T^{2} \)
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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

−13.12433836863149865100830434158, −12.08344022708005250191569939388, −10.13655837055508702847605005087, −9.521612591449673733134738064325, −8.836008506415468411963068545483, −7.05728864927297236006990901245, −5.63183645810342637622574228926, −4.66930642225311838294725724891, −2.59860377209100763877786397351, −1.65140128681451101786827233322, 0.68934821668634924650270922333, 2.49475120220288297455094144854, 3.26894236525217297144101253918, 5.66183222732066221459576548142, 6.44428325191071232211418117265, 7.76108254908827597458720901481, 9.197801466166240683729803067761, 10.28468695851973369598248366774, 10.86311339376152360887294654435, 12.99463900058995132239020645376

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