Properties

Label 2-2e6-16.13-c3-0-0
Degree $2$
Conductor $64$
Sign $-0.541 - 0.840i$
Analytic cond. $3.77612$
Root an. cond. $1.94322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.49 − 5.49i)3-s + (−4.66 + 4.66i)5-s + 24.8i·7-s + 33.4i·9-s + (−22.3 + 22.3i)11-s + (−11.2 − 11.2i)13-s + 51.2·15-s − 88.4·17-s + (−37.8 − 37.8i)19-s + (136. − 136. i)21-s − 48.1i·23-s + 81.4i·25-s + (35.2 − 35.2i)27-s + (10.4 + 10.4i)29-s + 96.9·31-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)3-s + (−0.417 + 0.417i)5-s + 1.34i·7-s + 1.23i·9-s + (−0.612 + 0.612i)11-s + (−0.240 − 0.240i)13-s + 0.882·15-s − 1.26·17-s + (−0.456 − 0.456i)19-s + (1.42 − 1.42i)21-s − 0.436i·23-s + 0.651i·25-s + (0.251 − 0.251i)27-s + (0.0668 + 0.0668i)29-s + 0.561·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(3.77612\)
Root analytic conductor: \(1.94322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :3/2),\ -0.541 - 0.840i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.141744 + 0.259805i\)
\(L(\frac12)\) \(\approx\) \(0.141744 + 0.259805i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.49 + 5.49i)T + 27iT^{2} \)
5 \( 1 + (4.66 - 4.66i)T - 125iT^{2} \)
7 \( 1 - 24.8iT - 343T^{2} \)
11 \( 1 + (22.3 - 22.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (11.2 + 11.2i)T + 2.19e3iT^{2} \)
17 \( 1 + 88.4T + 4.91e3T^{2} \)
19 \( 1 + (37.8 + 37.8i)T + 6.85e3iT^{2} \)
23 \( 1 + 48.1iT - 1.21e4T^{2} \)
29 \( 1 + (-10.4 - 10.4i)T + 2.43e4iT^{2} \)
31 \( 1 - 96.9T + 2.97e4T^{2} \)
37 \( 1 + (163. - 163. i)T - 5.06e4iT^{2} \)
41 \( 1 + 360. iT - 6.89e4T^{2} \)
43 \( 1 + (-100. + 100. i)T - 7.95e4iT^{2} \)
47 \( 1 + 220.T + 1.03e5T^{2} \)
53 \( 1 + (175. - 175. i)T - 1.48e5iT^{2} \)
59 \( 1 + (405. - 405. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-664. - 664. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-107. - 107. i)T + 3.00e5iT^{2} \)
71 \( 1 + 215. iT - 3.57e5T^{2} \)
73 \( 1 + 668. iT - 3.89e5T^{2} \)
79 \( 1 - 822.T + 4.93e5T^{2} \)
83 \( 1 + (326. + 326. i)T + 5.71e5iT^{2} \)
89 \( 1 + 262. iT - 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−15.06024120481589020645124064270, −13.30310534410837565698214089828, −12.41758293544252354880770483530, −11.67796291827843450976752553824, −10.63372963680282454274324311090, −8.816889072631537140391828041629, −7.36715510546635874900084694884, −6.33520672240567997980768824371, −5.07392092569528286742347654677, −2.32315365300270425872717217964, 0.21883519007537294758173720720, 3.98641014341787647890747659331, 4.91204636899722761909872859586, 6.50278905980661357821655611895, 8.136041754213619151200383658665, 9.770415910267995599806886660579, 10.75182056216007347660158727906, 11.41340954701605657510550209120, 12.85686257611817355518347225257, 14.06582958568053051250019358591

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