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Properties

Label 2-2e6-16.5-c11-0-16
Degree $2$
Conductor $64$
Sign $0.0586 + 0.998i$
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  + (179. − 179. i)3-s + (2.96e3 + 2.96e3i)5-s + 5.17e3i·7-s + 1.12e5i·9-s + (−6.94e5 − 6.94e5i)11-s + (5.24e5 − 5.24e5i)13-s + 1.06e6·15-s + 1.82e6·17-s + (6.01e5 − 6.01e5i)19-s + (9.30e5 + 9.30e5i)21-s − 2.32e7i·23-s − 3.12e7i·25-s + (5.20e7 + 5.20e7i)27-s + (1.89e5 − 1.89e5i)29-s + 2.48e8·31-s + ⋯
L(s)  = 1  + (0.426 − 0.426i)3-s + (0.423 + 0.423i)5-s + 0.116i·7-s + 0.635i·9-s + (−1.30 − 1.30i)11-s + (0.391 − 0.391i)13-s + 0.361·15-s + 0.311·17-s + (0.0557 − 0.0557i)19-s + (0.0497 + 0.0497i)21-s − 0.754i·23-s − 0.640i·25-s + (0.698 + 0.698i)27-s + (0.00171 − 0.00171i)29-s + 1.56·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0586 + 0.998i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.0586 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.0586 + 0.998i$
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: \((2,\ 64,\ (\ :11/2),\ 0.0586 + 0.998i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.48593 - 1.40123i\)
\(L(\frac12)\) \(\approx\) \(1.48593 - 1.40123i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-179. + 179. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-2.96e3 - 2.96e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 5.17e3iT - 1.97e9T^{2} \)
11 \( 1 + (6.94e5 + 6.94e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-5.24e5 + 5.24e5i)T - 1.79e12iT^{2} \)
17 \( 1 - 1.82e6T + 3.42e13T^{2} \)
19 \( 1 + (-6.01e5 + 6.01e5i)T - 1.16e14iT^{2} \)
23 \( 1 + 2.32e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.89e5 + 1.89e5i)T - 1.22e16iT^{2} \)
31 \( 1 - 2.48e8T + 2.54e16T^{2} \)
37 \( 1 + (4.30e8 + 4.30e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 1.02e8iT - 5.50e17T^{2} \)
43 \( 1 + (-6.91e7 - 6.91e7i)T + 9.29e17iT^{2} \)
47 \( 1 + 2.50e8T + 2.47e18T^{2} \)
53 \( 1 + (1.76e9 + 1.76e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (3.82e9 + 3.82e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-7.80e9 + 7.80e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-6.53e9 + 6.53e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.53e10iT - 2.31e20T^{2} \)
73 \( 1 + 2.49e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.46e10T + 7.47e20T^{2} \)
83 \( 1 + (-2.00e10 + 2.00e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 3.72e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.06e11T + 7.15e21T^{2} \)
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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

−12.57230388267231467739197673366, −10.96706027654197550442878121907, −10.27100792245354177064949841725, −8.554750578697547405882667028522, −7.83035823988528629941547549817, −6.31731377635457556171647957457, −5.15315034938036618294537582818, −3.14355979394702174638097519972, −2.22749361804083218435937309490, −0.52461762515479888771235916030, 1.30500288598043209490684241489, 2.78412263583940911012416371931, 4.25618317076531866878733635392, 5.45587278584638457767996045548, 7.02999812506325172843676081189, 8.373014370845006280408288812327, 9.569886237772324508179259742526, 10.25389763360402831787531719060, 11.86855087439206574956483268424, 12.94212969152415126659173693311

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