↑

Properties

Label 2-2e6-16.13-c11-0-7
Degree $2$
Conductor $64$
Sign $-0.896 + 0.442i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−585. − 585. i)3-s + (−6.85e3 + 6.85e3i)5-s + 1.99e4i·7-s + 5.08e5i·9-s + (9.64e4 − 9.64e4i)11-s + (−6.74e5 − 6.74e5i)13-s + 8.03e6·15-s + 3.70e6·17-s + (2.61e6 + 2.61e6i)19-s + (1.16e7 − 1.16e7i)21-s + 2.97e7i·23-s − 4.52e7i·25-s + (1.94e8 − 1.94e8i)27-s + (4.14e7 + 4.14e7i)29-s − 6.07e7·31-s + ⋯
L(s)  = 1  + (−1.39 − 1.39i)3-s + (−0.981 + 0.981i)5-s + 0.447i·7-s + 2.87i·9-s + (0.180 − 0.180i)11-s + (−0.503 − 0.503i)13-s + 2.73·15-s + 0.633·17-s + (0.242 + 0.242i)19-s + (0.623 − 0.623i)21-s + 0.964i·23-s − 0.927i·25-s + (2.60 − 2.60i)27-s + (0.375 + 0.375i)29-s − 0.381·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ -0.896 + 0.442i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0213881 - 0.0917575i\)
\(L(\frac12)\) \(\approx\) \(0.0213881 - 0.0917575i\)
\(L(\frac{13}{2})\) 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 \( 1 + (585. + 585. i)T + 1.77e5iT^{2} \)
5 \( 1 + (6.85e3 - 6.85e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 1.99e4iT - 1.97e9T^{2} \)
11 \( 1 + (-9.64e4 + 9.64e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (6.74e5 + 6.74e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 3.70e6T + 3.42e13T^{2} \)
19 \( 1 + (-2.61e6 - 2.61e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 2.97e7iT - 9.52e14T^{2} \)
29 \( 1 + (-4.14e7 - 4.14e7i)T + 1.22e16iT^{2} \)
31 \( 1 + 6.07e7T + 2.54e16T^{2} \)
37 \( 1 + (4.37e8 - 4.37e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.20e9iT - 5.50e17T^{2} \)
43 \( 1 + (-1.45e8 + 1.45e8i)T - 9.29e17iT^{2} \)
47 \( 1 + 1.52e9T + 2.47e18T^{2} \)
53 \( 1 + (2.52e9 - 2.52e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-4.54e9 + 4.54e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-9.96e7 - 9.96e7i)T + 4.35e19iT^{2} \)
67 \( 1 + (7.32e9 + 7.32e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.31e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.59e9iT - 3.13e20T^{2} \)
79 \( 1 - 4.08e9T + 7.47e20T^{2} \)
83 \( 1 + (3.03e10 + 3.03e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 5.68e10iT - 2.77e21T^{2} \)
97 \( 1 + 5.73e10T + 7.15e21T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.91663552560947790725489634462, −11.41117838543712723619770080420, −10.32480545626006928298379383524, −7.996304257668114605380503033495, −7.27707280096224589578973178950, −6.26342877360735430378650944666, −5.12364837367845124593340414450, −3.04359515936620168051659811962, −1.43549955127425335282576366287, −0.04754540626843157066939217710, 0.78248376177863601027608068413, 3.79198434088868229108484215180, 4.51631676062435275664615894808, 5.45288619019586296196860958849, 7.00560237039823651045747573638, 8.757100400629858597155613593558, 9.845621618229043715937681819651, 10.88051393236340330031174245501, 11.91899280063485426830267476574, 12.45170529443942388078500174998

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