Properties

Label 2-84-7.5-c8-0-2
Degree $2$
Conductor $84$
Sign $-0.925 - 0.377i$
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 + (790. + 456. i)5-s + (−2.31e3 + 623. i)7-s + (1.09e3 − 1.89e3i)9-s + (4.31e3 + 7.46e3i)11-s + 3.71e4i·13-s − 4.26e4·15-s + (4.37e4 − 2.52e4i)17-s + (−1.28e4 − 7.43e3i)19-s + (7.93e4 − 7.94e4i)21-s + (2.18e5 − 3.79e5i)23-s + (2.20e5 + 3.82e5i)25-s + 1.02e5i·27-s − 1.55e5·29-s + (−1.46e6 + 8.44e5i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (1.26 + 0.729i)5-s + (−0.965 + 0.259i)7-s + (0.166 − 0.288i)9-s + (0.294 + 0.510i)11-s + 1.29i·13-s − 0.842·15-s + (0.523 − 0.302i)17-s + (−0.0988 − 0.0570i)19-s + (0.407 − 0.408i)21-s + (0.782 − 1.35i)23-s + (0.565 + 0.979i)25-s + 0.192i·27-s − 0.219·29-s + (−1.58 + 0.914i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.925 - 0.377i$
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: \((2,\ 84,\ (\ :4),\ -0.925 - 0.377i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.227343 + 1.15872i\)
\(L(\frac12)\) \(\approx\) \(0.227343 + 1.15872i\)
\(L(5)\) 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 \( 1 + (2.31e3 - 623. i)T \)
good5 \( 1 + (-790. - 456. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-4.31e3 - 7.46e3i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 3.71e4iT - 8.15e8T^{2} \)
17 \( 1 + (-4.37e4 + 2.52e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.28e4 + 7.43e3i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-2.18e5 + 3.79e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.55e5T + 5.00e11T^{2} \)
31 \( 1 + (1.46e6 - 8.44e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (1.15e6 - 2.00e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 4.20e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.43e6T + 1.16e13T^{2} \)
47 \( 1 + (4.11e6 + 2.37e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (1.47e6 + 2.55e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (1.62e7 - 9.37e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.45e7 + 8.40e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (9.81e6 + 1.70e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 3.56e7T + 6.45e14T^{2} \)
73 \( 1 + (1.35e7 - 7.82e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-9.07e6 + 1.57e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 1.15e7iT - 2.25e15T^{2} \)
89 \( 1 + (2.46e7 + 1.42e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 1.52e8iT - 7.83e15T^{2} \)
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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

−13.10031931966560359541159954116, −12.03238293358203701204548894199, −10.74039682821059371990103985182, −9.808150930434739889525841460673, −9.113002244678842784494323401026, −6.82220381553926854099654567902, −6.34936159191507341547238664847, −4.92956038829158298039703272930, −3.16668684553218597511529443172, −1.74733425070289031588855909583, 0.36481598643142385448790900247, 1.60361528444397553611550627534, 3.39335041991840624474231740190, 5.42605870301322051672484180069, 5.95466449136612744799560326573, 7.42187939689417209300338217418, 9.019801173427792739356595808925, 9.899715508478336012390066260616, 10.94777210649469765797019174336, 12.55066984138350763374547727033

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