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Properties

Label 2-84-28.27-c11-0-31
Degree $2$
Conductor $84$
Sign $-0.735 - 0.677i$
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  + (10.0 + 44.1i)2-s − 243·3-s + (−1.84e3 + 885. i)4-s − 669. i·5-s + (−2.43e3 − 1.07e4i)6-s + (1.64e4 + 4.12e4i)7-s + (−5.75e4 − 7.26e4i)8-s + 5.90e4·9-s + (2.95e4 − 6.71e3i)10-s + 2.27e5i·11-s + (4.48e5 − 2.15e5i)12-s − 6.11e5i·13-s + (−1.65e6 + 1.14e6i)14-s + 1.62e5i·15-s + (2.62e6 − 3.26e6i)16-s − 3.10e6i·17-s + ⋯
L(s)  = 1  + (0.221 + 0.975i)2-s − 0.577·3-s + (−0.901 + 0.432i)4-s − 0.0958i·5-s + (−0.127 − 0.562i)6-s + (0.370 + 0.928i)7-s + (−0.621 − 0.783i)8-s + 0.333·9-s + (0.0934 − 0.0212i)10-s + 0.425i·11-s + (0.520 − 0.249i)12-s − 0.456i·13-s + (−0.823 + 0.567i)14-s + 0.0553i·15-s + (0.626 − 0.779i)16-s − 0.530i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.735 - 0.677i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.625942 + 1.60390i\)
\(L(\frac12)\) \(\approx\) \(0.625942 + 1.60390i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10.0 - 44.1i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-1.64e4 - 4.12e4i)T \)
good5 \( 1 + 669. iT - 4.88e7T^{2} \)
11 \( 1 - 2.27e5iT - 2.85e11T^{2} \)
13 \( 1 + 6.11e5iT - 1.79e12T^{2} \)
17 \( 1 + 3.10e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.78e7T + 1.16e14T^{2} \)
23 \( 1 + 2.18e5iT - 9.52e14T^{2} \)
29 \( 1 - 8.06e7T + 1.22e16T^{2} \)
31 \( 1 - 1.50e8T + 2.54e16T^{2} \)
37 \( 1 + 5.67e8T + 1.77e17T^{2} \)
41 \( 1 - 1.05e9iT - 5.50e17T^{2} \)
43 \( 1 + 4.42e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.81e8T + 2.47e18T^{2} \)
53 \( 1 + 1.07e9T + 9.26e18T^{2} \)
59 \( 1 - 3.35e9T + 3.01e19T^{2} \)
61 \( 1 - 2.15e9iT - 4.35e19T^{2} \)
67 \( 1 - 9.10e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.41e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.01e9iT - 3.13e20T^{2} \)
79 \( 1 + 3.09e10iT - 7.47e20T^{2} \)
83 \( 1 + 4.63e10T + 1.28e21T^{2} \)
89 \( 1 - 2.69e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.27e11iT - 7.15e21T^{2} \)
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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

−12.40830099198093897235460780894, −11.70740922764957293823663594618, −10.04545451507828180436551396851, −8.943084308341754815070271849818, −7.79663231928191043031419041185, −6.65653175145339291193963115048, −5.42976105182450418827594767372, −4.77904762394165431115730718176, −3.00823033098461107211658577759, −0.976396781873640026920276845406, 0.57435059647198666948162557951, 1.48729734337601232347942146584, 3.19774264254964171189493555607, 4.38306117673223972437891879439, 5.45852235589814574014786400179, 6.94868996200319043914756883213, 8.442294824111670285202228227883, 9.835214158625512350532471945651, 10.70192703757719423835731852941, 11.52268621466417479919953956522

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