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Properties

Label 2-2e6-8.5-c21-0-39
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  − 3.46e4i·3-s − 3.82e7i·5-s + 1.41e9·7-s + 9.25e9·9-s − 3.55e10i·11-s − 7.27e11i·13-s − 1.32e12·15-s + 5.23e12·17-s − 2.49e13i·19-s − 4.92e13i·21-s − 1.89e14·23-s − 9.84e14·25-s − 6.83e14i·27-s + 1.26e15i·29-s − 7.63e15·31-s + ⋯
L(s)  = 1  − 0.339i·3-s − 1.75i·5-s + 1.89·7-s + 0.884·9-s − 0.413i·11-s − 1.46i·13-s − 0.593·15-s + 0.629·17-s − 0.933i·19-s − 0.644i·21-s − 0.954·23-s − 2.06·25-s − 0.639i·27-s + 0.558i·29-s − 1.67·31-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

\(L(11)\) \(\approx\) \(3.144112337\)
\(L(\frac12)\) \(\approx\) \(3.144112337\)
\(L(\frac{23}{2})\) 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 \( 1 + 3.46e4iT - 1.04e10T^{2} \)
5 \( 1 + 3.82e7iT - 4.76e14T^{2} \)
7 \( 1 - 1.41e9T + 5.58e17T^{2} \)
11 \( 1 + 3.55e10iT - 7.40e21T^{2} \)
13 \( 1 + 7.27e11iT - 2.47e23T^{2} \)
17 \( 1 - 5.23e12T + 6.90e25T^{2} \)
19 \( 1 + 2.49e13iT - 7.14e26T^{2} \)
23 \( 1 + 1.89e14T + 3.94e28T^{2} \)
29 \( 1 - 1.26e15iT - 5.13e30T^{2} \)
31 \( 1 + 7.63e15T + 2.08e31T^{2} \)
37 \( 1 - 5.53e15iT - 8.55e32T^{2} \)
41 \( 1 - 1.10e17T + 7.38e33T^{2} \)
43 \( 1 + 6.92e16iT - 2.00e34T^{2} \)
47 \( 1 + 6.41e16T + 1.30e35T^{2} \)
53 \( 1 + 1.93e18iT - 1.62e36T^{2} \)
59 \( 1 + 1.53e18iT - 1.54e37T^{2} \)
61 \( 1 - 8.24e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.84e18iT - 2.22e38T^{2} \)
71 \( 1 + 1.43e19T + 7.52e38T^{2} \)
73 \( 1 + 1.03e19T + 1.34e39T^{2} \)
79 \( 1 - 1.00e20T + 7.08e39T^{2} \)
83 \( 1 - 1.29e20iT - 1.99e40T^{2} \)
89 \( 1 - 3.30e20T + 8.65e40T^{2} \)
97 \( 1 + 1.51e20T + 5.27e41T^{2} \)
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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

−10.49461578828540911610919308488, −9.070971682013081656841151557149, −8.130234832158436065873686702300, −7.59247689372926377062708710276, −5.52886186440027397777317206265, −5.00065734242448976635019924125, −3.97760912205447910304340550196, −2.02040618733643599167577181629, −1.17470781180174987375156101858, −0.58908752642754836878205107985, 1.60244035785028750252982701315, 2.12022332769107858219376158632, 3.75866380234845496571227991042, 4.50019171095742170322257274013, 5.95526585901772156859774216044, 7.25375228191232794958717879971, 7.79502295084914019981891731448, 9.479371592339982427400408529036, 10.52597237424359813675133237783, 11.20359459022385643467819361440

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