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Properties

Label 2-2e7-16.5-c11-0-20
Degree $2$
Conductor $128$
Sign $-0.243 + 0.969i$
Analytic cond. $98.3479$
Root an. cond. $9.91705$
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  + (199. − 199. i)3-s + (−8.93e3 − 8.93e3i)5-s + 5.56e4i·7-s + 9.75e4i·9-s + (1.46e5 + 1.46e5i)11-s + (−1.33e6 + 1.33e6i)13-s − 3.56e6·15-s − 2.92e6·17-s + (1.11e7 − 1.11e7i)19-s + (1.11e7 + 1.11e7i)21-s − 2.12e7i·23-s + 1.10e8i·25-s + (5.48e7 + 5.48e7i)27-s + (1.31e7 − 1.31e7i)29-s + 1.84e8·31-s + ⋯
L(s)  = 1  + (0.474 − 0.474i)3-s + (−1.27 − 1.27i)5-s + 1.25i·7-s + 0.550i·9-s + (0.273 + 0.273i)11-s + (−0.994 + 0.994i)13-s − 1.21·15-s − 0.499·17-s + (1.03 − 1.03i)19-s + (0.593 + 0.593i)21-s − 0.688i·23-s + 2.27i·25-s + (0.735 + 0.735i)27-s + (0.119 − 0.119i)29-s + 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.243 + 0.969i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ -0.243 + 0.969i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.131435895\)
\(L(\frac12)\) \(\approx\) \(1.131435895\)
\(L(\frac{13}{2})\) 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 \( 1 + (-199. + 199. i)T - 1.77e5iT^{2} \)
5 \( 1 + (8.93e3 + 8.93e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 5.56e4iT - 1.97e9T^{2} \)
11 \( 1 + (-1.46e5 - 1.46e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (1.33e6 - 1.33e6i)T - 1.79e12iT^{2} \)
17 \( 1 + 2.92e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.11e7 + 1.11e7i)T - 1.16e14iT^{2} \)
23 \( 1 + 2.12e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.31e7 + 1.31e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 1.84e8T + 2.54e16T^{2} \)
37 \( 1 + (6.58e7 + 6.58e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 7.48e8iT - 5.50e17T^{2} \)
43 \( 1 + (-3.10e8 - 3.10e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.85e9T + 2.47e18T^{2} \)
53 \( 1 + (3.45e9 + 3.45e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (8.57e8 + 8.57e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (6.80e9 - 6.80e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-5.85e9 + 5.85e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.15e8iT - 2.31e20T^{2} \)
73 \( 1 - 1.23e9iT - 3.13e20T^{2} \)
79 \( 1 + 1.66e9T + 7.47e20T^{2} \)
83 \( 1 + (-1.21e10 + 1.21e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 3.35e10iT - 2.77e21T^{2} \)
97 \( 1 - 2.49e10T + 7.15e21T^{2} \)
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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

−11.37142294811057556898443293415, −9.416384617022618867962837770570, −8.738632742048952375249076037664, −7.924866267615400824109313346783, −6.90460866787938808153137585473, −5.08878339518061894281533086453, −4.48562058456057284920150979780, −2.79416409006086120508254876279, −1.71217354537975909274947653207, −0.30322432045776928289782858737, 0.841196131272858351145203474947, 3.02567597820044019101382132300, 3.50713330488692662583247055192, 4.50876177491651133903865674866, 6.43442667619619079712770646693, 7.45610824299902215163536351352, 8.046422691437087404350274480707, 9.735025820291814869429412647158, 10.41807460702452168740289159919, 11.39506409733346333176551979328

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