↑

Properties

Label 2-2e7-16.5-c11-0-19
Degree $2$
Conductor $128$
Sign $0.997 + 0.0705i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0705i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.997 + 0.0705i$
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: \((2,\ 128,\ (\ :11/2),\ 0.997 + 0.0705i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.993337353\)
\(L(\frac12)\) \(\approx\) \(3.993337353\)
\(L(\frac{13}{2})\) 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 \( 1 - 1.08e10T + 7.15e21T^{2} \)
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

−11.61794950636147463795363095134, −9.641985813114326716501981053251, −9.182947855791832477022481237108, −7.962386438048243968687239610396, −6.96672255568562259102021897433, −6.29116589889588611954171149397, −4.40802721996509348209157291647, −2.70817596192320018836198648718, −2.27169455034151869779916458803, −1.13723990371405910980839251803, 0.789834825476094763526370794358, 2.36426172932670394764425548032, 3.46466312006672771796911625242, 4.37671859260701867435330970126, 5.45639031396283411276076194176, 7.19704336489479105829981987806, 8.504625757961770098817044153805, 9.065937863038504862482543327927, 10.14745137988753356556702039777, 10.71721814209785926341450433548

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