Properties

Label 2-84-21.20-c11-0-7
Degree $2$
Conductor $84$
Sign $-0.752 + 0.658i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (138. + 397. i)3-s − 8.83e3·5-s + (1.66e4 + 4.12e4i)7-s + (−1.38e5 + 1.09e5i)9-s + 6.74e5i·11-s + 1.36e6i·13-s + (−1.22e6 − 3.51e6i)15-s + 7.92e6·17-s + 8.91e6i·19-s + (−1.40e7 + 1.23e7i)21-s + 2.25e7i·23-s + 2.91e7·25-s + (−6.28e7 − 4.00e7i)27-s − 8.14e7i·29-s + 2.35e8i·31-s + ⋯
L(s)  = 1  + (0.328 + 0.944i)3-s − 1.26·5-s + (0.374 + 0.927i)7-s + (−0.784 + 0.620i)9-s + 1.26i·11-s + 1.01i·13-s + (−0.414 − 1.19i)15-s + 1.35·17-s + 0.826i·19-s + (−0.752 + 0.658i)21-s + 0.730i·23-s + 0.597·25-s + (−0.843 − 0.537i)27-s − 0.737i·29-s + 1.47i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.752 + 0.658i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.752 + 0.658i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.463550 - 1.23414i\)
\(L(\frac12)\) \(\approx\) \(0.463550 - 1.23414i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-138. - 397. i)T \)
7 \( 1 + (-1.66e4 - 4.12e4i)T \)
good5 \( 1 + 8.83e3T + 4.88e7T^{2} \)
11 \( 1 - 6.74e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.36e6iT - 1.79e12T^{2} \)
17 \( 1 - 7.92e6T + 3.42e13T^{2} \)
19 \( 1 - 8.91e6iT - 1.16e14T^{2} \)
23 \( 1 - 2.25e7iT - 9.52e14T^{2} \)
29 \( 1 + 8.14e7iT - 1.22e16T^{2} \)
31 \( 1 - 2.35e8iT - 2.54e16T^{2} \)
37 \( 1 + 5.33e8T + 1.77e17T^{2} \)
41 \( 1 + 9.52e8T + 5.50e17T^{2} \)
43 \( 1 - 1.43e9T + 9.29e17T^{2} \)
47 \( 1 - 2.52e9T + 2.47e18T^{2} \)
53 \( 1 + 2.77e9iT - 9.26e18T^{2} \)
59 \( 1 - 6.78e9T + 3.01e19T^{2} \)
61 \( 1 - 4.33e8iT - 4.35e19T^{2} \)
67 \( 1 + 1.56e10T + 1.22e20T^{2} \)
71 \( 1 + 6.69e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.11e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.95e10T + 7.47e20T^{2} \)
83 \( 1 - 5.39e10T + 1.28e21T^{2} \)
89 \( 1 + 3.45e10T + 2.77e21T^{2} \)
97 \( 1 - 2.62e10iT - 7.15e21T^{2} \)
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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

−12.11114047904422339529506148555, −11.88407055960520591123115408256, −10.44570669069642818785878842122, −9.359507784933126330464320804791, −8.323466213418343315814177374267, −7.34713377677220003872398219306, −5.44325511506097094010682484049, −4.36584734687914901884034373771, −3.38566280963519058190710029489, −1.85372002098784419792761246856, 0.38955853444116647208019924700, 0.952171070932212642805570500421, 2.94804278324753577721562102877, 3.87905883608929539313537225323, 5.62441183888518674561151539498, 7.16401039666698181587891532204, 7.86515767963819902644769114395, 8.653880835108374562473725636846, 10.55184283602737811292994747804, 11.47459843896244042708662581259

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