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Properties

Label 2-2e6-16.13-c11-0-6
Degree $2$
Conductor $64$
Sign $-0.968 - 0.248i$
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  + (522. + 522. i)3-s + (−3.20e3 + 3.20e3i)5-s + 3.19e4i·7-s + 3.69e5i·9-s + (4.06e5 − 4.06e5i)11-s + (1.12e6 + 1.12e6i)13-s − 3.34e6·15-s − 5.55e6·17-s + (7.81e6 + 7.81e6i)19-s + (−1.67e7 + 1.67e7i)21-s + 3.53e7i·23-s + 2.83e7i·25-s + (−1.00e8 + 1.00e8i)27-s + (−1.32e8 − 1.32e8i)29-s − 7.64e7·31-s + ⋯
L(s)  = 1  + (1.24 + 1.24i)3-s + (−0.458 + 0.458i)5-s + 0.719i·7-s + 2.08i·9-s + (0.761 − 0.761i)11-s + (0.840 + 0.840i)13-s − 1.13·15-s − 0.948·17-s + (0.723 + 0.723i)19-s + (−0.893 + 0.893i)21-s + 1.14i·23-s + 0.579i·25-s + (−1.34 + 1.34i)27-s + (−1.20 − 1.20i)29-s − 0.479·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.968 - 0.248i$
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: \((2,\ 64,\ (\ :11/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.344341 + 2.72443i\)
\(L(\frac12)\) \(\approx\) \(0.344341 + 2.72443i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-522. - 522. i)T + 1.77e5iT^{2} \)
5 \( 1 + (3.20e3 - 3.20e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 3.19e4iT - 1.97e9T^{2} \)
11 \( 1 + (-4.06e5 + 4.06e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-1.12e6 - 1.12e6i)T + 1.79e12iT^{2} \)
17 \( 1 + 5.55e6T + 3.42e13T^{2} \)
19 \( 1 + (-7.81e6 - 7.81e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 3.53e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.32e8 + 1.32e8i)T + 1.22e16iT^{2} \)
31 \( 1 + 7.64e7T + 2.54e16T^{2} \)
37 \( 1 + (6.43e7 - 6.43e7i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.19e9iT - 5.50e17T^{2} \)
43 \( 1 + (-1.26e9 + 1.26e9i)T - 9.29e17iT^{2} \)
47 \( 1 + 5.29e8T + 2.47e18T^{2} \)
53 \( 1 + (3.02e9 - 3.02e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (2.78e9 - 2.78e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (3.36e9 + 3.36e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-1.24e9 - 1.24e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.89e9iT - 2.31e20T^{2} \)
73 \( 1 + 7.84e9iT - 3.13e20T^{2} \)
79 \( 1 + 1.00e10T + 7.47e20T^{2} \)
83 \( 1 + (-4.69e9 - 4.69e9i)T + 1.28e21iT^{2} \)
89 \( 1 + 2.92e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.08e10T + 7.15e21T^{2} \)
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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

−13.60545748766870063828176492182, −11.66800916839218523901304046108, −10.82824494650271652302717201225, −9.273237828804212333046367263925, −8.925798045942394770640923152160, −7.56687152130377208853428034222, −5.75662599097725807977609209687, −4.00620151576255380520846111986, −3.42560121732294951680211160775, −1.97244790991217790994440889069, 0.61887213730123858512350513694, 1.60004003021280476318852787463, 3.06080199002036565806282093530, 4.32656523551449893621269535587, 6.55004214146622896992813478862, 7.46487805261340275384034857488, 8.425452206472745307335532484592, 9.364351281691813409172746723363, 11.12242813581256902086102167436, 12.56773594174348143467120495981

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