Properties

Label 2-84-21.20-c11-0-11
Degree $2$
Conductor $84$
Sign $0.682 - 0.730i$
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  + (240. − 345. i)3-s − 113.·5-s + (4.40e4 + 6.36e3i)7-s + (−6.15e4 − 1.66e5i)9-s + 4.71e5i·11-s + 1.52e6i·13-s + (−2.72e4 + 3.91e4i)15-s + 1.63e6·17-s + 2.97e6i·19-s + (1.27e7 − 1.36e7i)21-s + 2.79e7i·23-s − 4.88e7·25-s + (−7.21e7 − 1.86e7i)27-s + 1.72e8i·29-s + 6.77e7i·31-s + ⋯
L(s)  = 1  + (0.571 − 0.820i)3-s − 0.0162·5-s + (0.989 + 0.143i)7-s + (−0.347 − 0.937i)9-s + 0.883i·11-s + 1.13i·13-s + (−0.00926 + 0.0133i)15-s + 0.278·17-s + 0.275i·19-s + (0.682 − 0.730i)21-s + 0.904i·23-s − 0.999·25-s + (−0.968 − 0.250i)27-s + 1.56i·29-s + 0.424i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.682 - 0.730i$
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: \((2,\ 84,\ (\ :11/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.21358 + 0.960836i\)
\(L(\frac12)\) \(\approx\) \(2.21358 + 0.960836i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-240. + 345. i)T \)
7 \( 1 + (-4.40e4 - 6.36e3i)T \)
good5 \( 1 + 113.T + 4.88e7T^{2} \)
11 \( 1 - 4.71e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.52e6iT - 1.79e12T^{2} \)
17 \( 1 - 1.63e6T + 3.42e13T^{2} \)
19 \( 1 - 2.97e6iT - 1.16e14T^{2} \)
23 \( 1 - 2.79e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.72e8iT - 1.22e16T^{2} \)
31 \( 1 - 6.77e7iT - 2.54e16T^{2} \)
37 \( 1 + 2.11e8T + 1.77e17T^{2} \)
41 \( 1 - 1.13e9T + 5.50e17T^{2} \)
43 \( 1 + 1.22e9T + 9.29e17T^{2} \)
47 \( 1 + 2.73e9T + 2.47e18T^{2} \)
53 \( 1 + 4.10e9iT - 9.26e18T^{2} \)
59 \( 1 - 9.33e9T + 3.01e19T^{2} \)
61 \( 1 - 9.27e9iT - 4.35e19T^{2} \)
67 \( 1 - 9.31e9T + 1.22e20T^{2} \)
71 \( 1 + 1.13e9iT - 2.31e20T^{2} \)
73 \( 1 - 1.18e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.57e9T + 7.47e20T^{2} \)
83 \( 1 - 3.83e10T + 1.28e21T^{2} \)
89 \( 1 - 4.88e10T + 2.77e21T^{2} \)
97 \( 1 - 1.23e11iT - 7.15e21T^{2} \)
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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

−12.12850392061473735160086846043, −11.40953639372210924331438647246, −9.773470714250000555229112053533, −8.689150750021338825042672945853, −7.65576460938972289225525905328, −6.73992321257894834197213270918, −5.17760992782570718892487749473, −3.73549669198767920091346668864, −2.08341608268798208000877026336, −1.38604522358929284425634662337, 0.54033576663538540849204812645, 2.25781998269837513103315019185, 3.52334715976568590971790553402, 4.72430531351554344494227952224, 5.84447290887853002706976508173, 7.84489414250270849760044854594, 8.385321801811815825978831121469, 9.747011829897564416442128155164, 10.74098738225083540167761143375, 11.59423589469467509627199080796

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