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Properties

Label 2-2e6-16.3-c4-0-0
Degree $2$
Conductor $64$
Sign $-0.974 + 0.225i$
Analytic cond. $6.61567$
Root an. cond. $2.57209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.91 + 3.91i)3-s + (4.72 − 4.72i)5-s − 45.3·7-s + 50.3i·9-s + (−110. − 110. i)11-s + (−157. − 157. i)13-s + 36.9i·15-s − 378.·17-s + (−203. + 203. i)19-s + (177. − 177. i)21-s + 740.·23-s + 580. i·25-s + (−514. − 514. i)27-s + (82.6 + 82.6i)29-s + 286. i·31-s + ⋯
L(s)  = 1  + (−0.434 + 0.434i)3-s + (0.188 − 0.188i)5-s − 0.925·7-s + 0.621i·9-s + (−0.910 − 0.910i)11-s + (−0.929 − 0.929i)13-s + 0.164i·15-s − 1.31·17-s + (−0.562 + 0.562i)19-s + (0.402 − 0.402i)21-s + 1.39·23-s + 0.928i·25-s + (−0.705 − 0.705i)27-s + (0.0982 + 0.0982i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(6.61567\)
Root analytic conductor: \(2.57209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :2),\ -0.974 + 0.225i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00259381 - 0.0226888i\)
\(L(\frac12)\) \(\approx\) \(0.00259381 - 0.0226888i\)
\(L(3)\) 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 \( 1 + (3.91 - 3.91i)T - 81iT^{2} \)
5 \( 1 + (-4.72 + 4.72i)T - 625iT^{2} \)
7 \( 1 + 45.3T + 2.40e3T^{2} \)
11 \( 1 + (110. + 110. i)T + 1.46e4iT^{2} \)
13 \( 1 + (157. + 157. i)T + 2.85e4iT^{2} \)
17 \( 1 + 378.T + 8.35e4T^{2} \)
19 \( 1 + (203. - 203. i)T - 1.30e5iT^{2} \)
23 \( 1 - 740.T + 2.79e5T^{2} \)
29 \( 1 + (-82.6 - 82.6i)T + 7.07e5iT^{2} \)
31 \( 1 - 286. iT - 9.23e5T^{2} \)
37 \( 1 + (-1.47e3 + 1.47e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + (-366. - 366. i)T + 3.41e6iT^{2} \)
47 \( 1 - 751. iT - 4.87e6T^{2} \)
53 \( 1 + (1.92e3 - 1.92e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (-1.35e3 - 1.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (1.83e3 + 1.83e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (2.20e3 - 2.20e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 8.97e3T + 2.54e7T^{2} \)
73 \( 1 + 9.35e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.03e3 + 1.03e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 5.17e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.53e3T + 8.85e7T^{2} \)
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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

−14.97153374797035574831878037188, −13.31677032100566431334966510336, −12.84488918076725430213935127781, −11.10355559453111450065765499475, −10.37925154354133386616411694317, −9.117186043515039332825897207665, −7.65090498242697163643414524836, −5.99883981006763380621235994648, −4.85284252720448864798262584640, −2.85398819724486607003885053364, 0.01216942244235272438964797042, 2.49173605378520222603565896834, 4.63406328877658163653628775024, 6.43780417329819131434334262458, 7.11864038300871306656563489415, 9.062482164401643774051357528723, 10.06915952799498489485941685361, 11.43947113252478466734956922569, 12.61846356860143111218985897576, 13.24661491109620734301919518883

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