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Properties

Label 2-2e6-16.5-c11-0-1
Degree $2$
Conductor $64$
Sign $-0.382 + 0.923i$
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  + (−457. + 457. i)3-s + (5.57e3 + 5.57e3i)5-s − 5.54e4i·7-s − 2.41e5i·9-s + (5.93e5 + 5.93e5i)11-s + (−3.20e5 + 3.20e5i)13-s − 5.09e6·15-s − 9.66e6·17-s + (−6.35e6 + 6.35e6i)19-s + (2.53e7 + 2.53e7i)21-s + 1.96e7i·23-s + 1.33e7i·25-s + (2.92e7 + 2.92e7i)27-s + (−7.64e7 + 7.64e7i)29-s + 1.66e8·31-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)3-s + (0.797 + 0.797i)5-s − 1.24i·7-s − 1.36i·9-s + (1.11 + 1.11i)11-s + (−0.239 + 0.239i)13-s − 1.73·15-s − 1.65·17-s + (−0.589 + 0.589i)19-s + (1.35 + 1.35i)21-s + 0.635i·23-s + 0.273i·25-s + (0.391 + 0.391i)27-s + (−0.692 + 0.692i)29-s + 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.382 + 0.923i$
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: \((2,\ 64,\ (\ :11/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0815288 - 0.122034i\)
\(L(\frac12)\) \(\approx\) \(0.0815288 - 0.122034i\)
\(L(\frac{13}{2})\) 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 \( 1 + (457. - 457. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-5.57e3 - 5.57e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 5.54e4iT - 1.97e9T^{2} \)
11 \( 1 + (-5.93e5 - 5.93e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (3.20e5 - 3.20e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 9.66e6T + 3.42e13T^{2} \)
19 \( 1 + (6.35e6 - 6.35e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 1.96e7iT - 9.52e14T^{2} \)
29 \( 1 + (7.64e7 - 7.64e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 1.66e8T + 2.54e16T^{2} \)
37 \( 1 + (-2.74e7 - 2.74e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 6.15e8iT - 5.50e17T^{2} \)
43 \( 1 + (8.26e8 + 8.26e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.20e9T + 2.47e18T^{2} \)
53 \( 1 + (-6.02e8 - 6.02e8i)T + 9.26e18iT^{2} \)
59 \( 1 + (3.88e9 + 3.88e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (2.43e9 - 2.43e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (8.86e9 - 8.86e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 2.93e10iT - 2.31e20T^{2} \)
73 \( 1 - 7.45e9iT - 3.13e20T^{2} \)
79 \( 1 - 2.31e10T + 7.47e20T^{2} \)
83 \( 1 + (4.40e9 - 4.40e9i)T - 1.28e21iT^{2} \)
89 \( 1 + 4.82e10iT - 2.77e21T^{2} \)
97 \( 1 - 9.19e10T + 7.15e21T^{2} \)
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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

−13.48746772271750574422118770274, −11.91237318208007443215470367270, −10.81366623268307908511375551364, −10.22481336199785716202869708128, −9.326035310595263714328172537305, −7.01360515996240783566782659092, −6.29804282869197039746870611203, −4.70686776136957886608470362461, −3.88702763998408254751958045825, −1.79550746404013283120898514603, 0.04644325864671063726745510355, 1.23024174809053592080208236193, 2.34609799396682262937011616893, 4.83729638275246290143413465654, 6.03063845062760782850197464642, 6.48826591763655389491654027206, 8.447616007036996652598378874548, 9.283057787253672973635879605180, 11.16228129060482492697866956323, 11.87415384883302129408386842608

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