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Properties

Label 2-2e6-8.5-c21-0-21
Degree $2$
Conductor $64$
Sign $-0.965 + 0.258i$
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  + 1.95e5i·3-s − 2.28e6i·5-s + 1.31e9·7-s − 2.75e10·9-s + 1.17e11i·11-s + 7.00e11i·13-s + 4.45e11·15-s + 1.39e12·17-s + 2.00e12i·19-s + 2.55e14i·21-s + 2.01e14·23-s + 4.71e14·25-s − 3.33e15i·27-s + 3.05e15i·29-s + 9.56e14·31-s + ⋯
L(s)  = 1  + 1.90i·3-s − 0.104i·5-s + 1.75·7-s − 2.63·9-s + 1.36i·11-s + 1.40i·13-s + 0.199·15-s + 0.168·17-s + 0.0750i·19-s + 3.34i·21-s + 1.01·23-s + 0.989·25-s − 3.11i·27-s + 1.34i·29-s + 0.209·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.965 + 0.258i$
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: \((2,\ 64,\ (\ :21/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.985653981\)
\(L(\frac12)\) \(\approx\) \(2.985653981\)
\(L(\frac{23}{2})\) 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 \( 1 - 1.95e5iT - 1.04e10T^{2} \)
5 \( 1 + 2.28e6iT - 4.76e14T^{2} \)
7 \( 1 - 1.31e9T + 5.58e17T^{2} \)
11 \( 1 - 1.17e11iT - 7.40e21T^{2} \)
13 \( 1 - 7.00e11iT - 2.47e23T^{2} \)
17 \( 1 - 1.39e12T + 6.90e25T^{2} \)
19 \( 1 - 2.00e12iT - 7.14e26T^{2} \)
23 \( 1 - 2.01e14T + 3.94e28T^{2} \)
29 \( 1 - 3.05e15iT - 5.13e30T^{2} \)
31 \( 1 - 9.56e14T + 2.08e31T^{2} \)
37 \( 1 + 2.17e16iT - 8.55e32T^{2} \)
41 \( 1 - 3.06e16T + 7.38e33T^{2} \)
43 \( 1 - 9.69e16iT - 2.00e34T^{2} \)
47 \( 1 - 5.67e17T + 1.30e35T^{2} \)
53 \( 1 - 1.26e18iT - 1.62e36T^{2} \)
59 \( 1 - 4.03e18iT - 1.54e37T^{2} \)
61 \( 1 - 4.90e18iT - 3.10e37T^{2} \)
67 \( 1 + 2.07e18iT - 2.22e38T^{2} \)
71 \( 1 - 7.65e18T + 7.52e38T^{2} \)
73 \( 1 - 1.76e19T + 1.34e39T^{2} \)
79 \( 1 + 6.20e19T + 7.08e39T^{2} \)
83 \( 1 - 1.43e19iT - 1.99e40T^{2} \)
89 \( 1 + 3.06e20T + 8.65e40T^{2} \)
97 \( 1 + 8.11e20T + 5.27e41T^{2} \)
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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

−11.16779872683264879228737254040, −10.53209358528799137319478149784, −9.283017758977510913757519873611, −8.699027041428808623967373124529, −7.24635571928869015612522962483, −5.38764275388179172154380565364, −4.61107482690908281590510984277, −4.18928186831479627573042759133, −2.60184479159675860775022871143, −1.36586056552703645896387111554, 0.67043823008999074569377534287, 0.971838305550648199416507069146, 2.17778132981985099412860354754, 3.11258515035210129014159790893, 5.16022955623105570229560956395, 5.96106868519907346566729722375, 7.22696031692172125254366235650, 8.143809036472590561273916126598, 8.518148732659991900317503570580, 10.88868828728889140588123102937

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