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Properties

Label 2-2e6-8.5-c21-0-22
Degree $2$
Conductor $64$
Sign $0.258 - 0.965i$
Analytic cond. $178.865$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.14e5i·3-s + 1.31e7i·5-s + 8.14e8·7-s − 2.68e9·9-s − 1.24e11i·11-s + 1.76e11i·13-s − 1.50e12·15-s − 3.32e12·17-s − 1.75e13i·19-s + 9.33e13i·21-s − 2.65e14·23-s + 3.04e14·25-s + 8.91e14i·27-s − 7.38e14i·29-s + 2.12e15·31-s + ⋯
L(s)  = 1  + 1.12i·3-s + 0.600i·5-s + 1.08·7-s − 0.256·9-s − 1.44i·11-s + 0.354i·13-s − 0.673·15-s − 0.400·17-s − 0.655i·19-s + 1.22i·21-s − 1.33·23-s + 0.639·25-s + 0.833i·27-s − 0.325i·29-s + 0.465·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(178.865\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :21/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.737269519\)
\(L(\frac12)\) \(\approx\) \(2.737269519\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 1.14e5iT - 1.04e10T^{2} \)
5 \( 1 - 1.31e7iT - 4.76e14T^{2} \)
7 \( 1 - 8.14e8T + 5.58e17T^{2} \)
11 \( 1 + 1.24e11iT - 7.40e21T^{2} \)
13 \( 1 - 1.76e11iT - 2.47e23T^{2} \)
17 \( 1 + 3.32e12T + 6.90e25T^{2} \)
19 \( 1 + 1.75e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.65e14T + 3.94e28T^{2} \)
29 \( 1 + 7.38e14iT - 5.13e30T^{2} \)
31 \( 1 - 2.12e15T + 2.08e31T^{2} \)
37 \( 1 - 3.13e16iT - 8.55e32T^{2} \)
41 \( 1 - 3.67e15T + 7.38e33T^{2} \)
43 \( 1 + 1.60e17iT - 2.00e34T^{2} \)
47 \( 1 - 4.23e17T + 1.30e35T^{2} \)
53 \( 1 + 1.51e18iT - 1.62e36T^{2} \)
59 \( 1 - 2.75e18iT - 1.54e37T^{2} \)
61 \( 1 - 1.67e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.15e19iT - 2.22e38T^{2} \)
71 \( 1 - 3.67e19T + 7.52e38T^{2} \)
73 \( 1 - 5.73e19T + 1.34e39T^{2} \)
79 \( 1 - 9.95e19T + 7.08e39T^{2} \)
83 \( 1 + 1.00e20iT - 1.99e40T^{2} \)
89 \( 1 - 3.88e20T + 8.65e40T^{2} \)
97 \( 1 - 4.38e20T + 5.27e41T^{2} \)
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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

−10.99775249018642494017216855533, −10.27169856690992732996543338585, −8.998483241429953271733354650887, −8.096955544440614279963347719809, −6.65782098059895853836199569256, −5.39097956195064236994585302987, −4.39413255232905941059914859886, −3.45726981808645420816333414781, −2.23043533239409454591534657496, −0.77770967151657575629962224043, 0.68409068211210063526264980477, 1.63687978635218600506350290685, 2.22314108836348009739785722176, 4.18093442738723025718166165600, 5.08431705915056204663976163070, 6.43223276901306503938159112179, 7.59665822645340569527213089967, 8.139212215933602035020195024708, 9.522643688273926632760293879473, 10.81178419699006604305774107702

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