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Properties

Label 2-2e6-16.5-c11-0-18
Degree $2$
Conductor $64$
Sign $-0.743 - 0.668i$
Analytic cond. $49.1739$
Root an. cond. $7.01241$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−255. + 255. i)3-s + (−4.21e3 − 4.21e3i)5-s − 8.03e4i·7-s + 4.61e4i·9-s + (−6.11e5 − 6.11e5i)11-s + (3.70e5 − 3.70e5i)13-s + 2.15e6·15-s − 7.25e5·17-s + (1.13e7 − 1.13e7i)19-s + (2.05e7 + 2.05e7i)21-s + 4.89e7i·23-s − 1.33e7i·25-s + (−5.71e7 − 5.71e7i)27-s + (4.98e7 − 4.98e7i)29-s − 4.20e7·31-s + ⋯
L(s)  = 1  + (−0.608 + 0.608i)3-s + (−0.603 − 0.603i)5-s − 1.80i·7-s + 0.260i·9-s + (−1.14 − 1.14i)11-s + (0.276 − 0.276i)13-s + 0.733·15-s − 0.123·17-s + (1.04 − 1.04i)19-s + (1.09 + 1.09i)21-s + 1.58i·23-s − 0.272i·25-s + (−0.766 − 0.766i)27-s + (0.451 − 0.451i)29-s − 0.263·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.743 - 0.668i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0725105 + 0.189210i\)
\(L(\frac12)\) \(\approx\) \(0.0725105 + 0.189210i\)
\(L(\frac{13}{2})\) 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 \( 1 + (255. - 255. i)T - 1.77e5iT^{2} \)
5 \( 1 + (4.21e3 + 4.21e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 8.03e4iT - 1.97e9T^{2} \)
11 \( 1 + (6.11e5 + 6.11e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-3.70e5 + 3.70e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 7.25e5T + 3.42e13T^{2} \)
19 \( 1 + (-1.13e7 + 1.13e7i)T - 1.16e14iT^{2} \)
23 \( 1 - 4.89e7iT - 9.52e14T^{2} \)
29 \( 1 + (-4.98e7 + 4.98e7i)T - 1.22e16iT^{2} \)
31 \( 1 + 4.20e7T + 2.54e16T^{2} \)
37 \( 1 + (1.05e8 + 1.05e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 4.49e8iT - 5.50e17T^{2} \)
43 \( 1 + (7.89e8 + 7.89e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.32e9T + 2.47e18T^{2} \)
53 \( 1 + (-1.74e9 - 1.74e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-6.71e8 - 6.71e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (8.30e8 - 8.30e8i)T - 4.35e19iT^{2} \)
67 \( 1 + (3.88e9 - 3.88e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 5.14e9iT - 2.31e20T^{2} \)
73 \( 1 + 4.42e9iT - 3.13e20T^{2} \)
79 \( 1 + 2.71e10T + 7.47e20T^{2} \)
83 \( 1 + (1.81e10 - 1.81e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 5.56e10iT - 2.77e21T^{2} \)
97 \( 1 - 2.61e10T + 7.15e21T^{2} \)
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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

−11.52204866662768305399332478925, −10.85125589977771460099567633042, −9.936032525995374345788871883211, −8.217916638319934936201268075256, −7.32528146223885219925251212255, −5.49365886064143958871820925519, −4.51575739882336326189408168749, −3.35709060725324638651802899677, −0.904526824023609249466544029619, −0.07544580796674734005923519827, 1.84347791462969529277223806648, 3.09027284584381552403286645685, 5.05960402778543700100493985238, 6.17221077864813426105326738715, 7.30479716497801361778373071056, 8.548530709272641630309235651277, 9.954380828965437132863731577091, 11.38447994684124372518018830553, 12.19996361676601134539155990987, 12.78312462761634516836534211133

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