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Properties

Label 2-2e6-16.11-c4-0-0
Degree $2$
Conductor $64$
Sign $-0.350 - 0.936i$
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  + (−11.5 − 11.5i)3-s + (−14.6 − 14.6i)5-s + 24.0·7-s + 184. i·9-s + (−61.7 + 61.7i)11-s + (−37.5 + 37.5i)13-s + 336. i·15-s + 96.8·17-s + (156. + 156. i)19-s + (−276. − 276. i)21-s − 959.·23-s − 198. i·25-s + (1.19e3 − 1.19e3i)27-s + (−350. + 350. i)29-s + 237. i·31-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)3-s + (−0.584 − 0.584i)5-s + 0.490·7-s + 2.27i·9-s + (−0.510 + 0.510i)11-s + (−0.222 + 0.222i)13-s + 1.49i·15-s + 0.335·17-s + (0.434 + 0.434i)19-s + (−0.627 − 0.627i)21-s − 1.81·23-s − 0.317i·25-s + (1.63 − 1.63i)27-s + (−0.416 + 0.416i)29-s + 0.247i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(6.61567\)
Root analytic conductor: \(2.57209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :2),\ -0.350 - 0.936i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00665223 + 0.00959304i\)
\(L(\frac12)\) \(\approx\) \(0.00665223 + 0.00959304i\)
\(L(3)\) 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 \( 1 + (11.5 + 11.5i)T + 81iT^{2} \)
5 \( 1 + (14.6 + 14.6i)T + 625iT^{2} \)
7 \( 1 - 24.0T + 2.40e3T^{2} \)
11 \( 1 + (61.7 - 61.7i)T - 1.46e4iT^{2} \)
13 \( 1 + (37.5 - 37.5i)T - 2.85e4iT^{2} \)
17 \( 1 - 96.8T + 8.35e4T^{2} \)
19 \( 1 + (-156. - 156. i)T + 1.30e5iT^{2} \)
23 \( 1 + 959.T + 2.79e5T^{2} \)
29 \( 1 + (350. - 350. i)T - 7.07e5iT^{2} \)
31 \( 1 - 237. iT - 9.23e5T^{2} \)
37 \( 1 + (560. + 560. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (206. - 206. i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.59e3iT - 4.87e6T^{2} \)
53 \( 1 + (2.23e3 + 2.23e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (2.35e3 - 2.35e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.44e3 - 4.44e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-3.99e3 - 3.99e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.92e3T + 2.54e7T^{2} \)
73 \( 1 - 2.65e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.79e3iT - 3.89e7T^{2} \)
83 \( 1 + (-228. - 228. i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.10e4T + 8.85e7T^{2} \)
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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

−14.28856147787597828382495115026, −13.02965413353687640304504229630, −12.15250728329032840684654135556, −11.60695509579489791060729638551, −10.25079516454004841660936045738, −8.156668407936483526274044257903, −7.36325453368452824099722624257, −5.91638746015601765324539003011, −4.71469591914218471105930839478, −1.65888233584283114018517948602, 0.00752060788616167915226867552, 3.58760025859545704206147106458, 4.94487532572644034896147858881, 6.09698771274741139626331011027, 7.80149735115584172789740408884, 9.575278406304969940219520924396, 10.64940392633402983196557547294, 11.32518752250132786005547433758, 12.23189564907527243809300790609, 14.11549620988590454781368445887

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