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Properties

Label 2-2e7-16.5-c11-0-16
Degree $2$
Conductor $128$
Sign $0.902 + 0.430i$
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  + (56.4 − 56.4i)3-s + (−6.26e3 − 6.26e3i)5-s − 3.29e4i·7-s + 1.70e5i·9-s + (1.51e5 + 1.51e5i)11-s + (1.05e6 − 1.05e6i)13-s − 7.06e5·15-s − 1.95e6·17-s + (−5.25e6 + 5.25e6i)19-s + (−1.86e6 − 1.86e6i)21-s + 4.62e7i·23-s + 2.95e7i·25-s + (1.96e7 + 1.96e7i)27-s + (−1.16e8 + 1.16e8i)29-s + 2.86e8·31-s + ⋯
L(s)  = 1  + (0.134 − 0.134i)3-s + (−0.895 − 0.895i)5-s − 0.740i·7-s + 0.964i·9-s + (0.284 + 0.284i)11-s + (0.785 − 0.785i)13-s − 0.240·15-s − 0.334·17-s + (−0.486 + 0.486i)19-s + (−0.0993 − 0.0993i)21-s + 1.49i·23-s + 0.605i·25-s + (0.263 + 0.263i)27-s + (−1.05 + 1.05i)29-s + 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.902 + 0.430i$
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: \((2,\ 128,\ (\ :11/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.727461727\)
\(L(\frac12)\) \(\approx\) \(1.727461727\)
\(L(\frac{13}{2})\) 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 \( 1 + (-56.4 + 56.4i)T - 1.77e5iT^{2} \)
5 \( 1 + (6.26e3 + 6.26e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 3.29e4iT - 1.97e9T^{2} \)
11 \( 1 + (-1.51e5 - 1.51e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-1.05e6 + 1.05e6i)T - 1.79e12iT^{2} \)
17 \( 1 + 1.95e6T + 3.42e13T^{2} \)
19 \( 1 + (5.25e6 - 5.25e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 4.62e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.16e8 - 1.16e8i)T - 1.22e16iT^{2} \)
31 \( 1 - 2.86e8T + 2.54e16T^{2} \)
37 \( 1 + (3.33e8 + 3.33e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 1.31e8iT - 5.50e17T^{2} \)
43 \( 1 + (5.60e8 + 5.60e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.10e8T + 2.47e18T^{2} \)
53 \( 1 + (-2.81e9 - 2.81e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-3.57e8 - 3.57e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (-6.15e9 + 6.15e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (8.27e9 - 8.27e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.51e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.11e10iT - 3.13e20T^{2} \)
79 \( 1 - 1.68e10T + 7.47e20T^{2} \)
83 \( 1 + (-1.04e10 + 1.04e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 3.00e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.53e11T + 7.15e21T^{2} \)
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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

−11.14014890977317523127539717259, −10.24948099052932481471462321976, −8.795021738663886715056250455119, −8.002592590350606586489115466335, −7.17050462519982852035422548031, −5.53282803257917272341635971502, −4.41628295464047078013650096956, −3.49162631132238448486655441937, −1.71396284086837602318297036889, −0.66281223785732193127155106348, 0.61492251827254981793519873172, 2.36092250771278734077943512262, 3.46661165390637623117248391388, 4.38462369665609565137246049714, 6.22525534061568675428896910055, 6.80114278325176706308121911752, 8.320249733369415604621972896345, 9.033076821075736072052204089165, 10.35156801172248770188044494097, 11.53226446631251605735071920318

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