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Properties

Label 2-2e7-16.5-c11-0-14
Degree $2$
Conductor $128$
Sign $0.923 - 0.382i$
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  + (−457. + 457. i)3-s + (−5.57e3 − 5.57e3i)5-s + 5.54e4i·7-s − 2.41e5i·9-s + (5.93e5 + 5.93e5i)11-s + (3.20e5 − 3.20e5i)13-s + 5.09e6·15-s − 9.66e6·17-s + (−6.35e6 + 6.35e6i)19-s + (−2.53e7 − 2.53e7i)21-s − 1.96e7i·23-s + 1.33e7i·25-s + (2.92e7 + 2.92e7i)27-s + (7.64e7 − 7.64e7i)29-s − 1.66e8·31-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)3-s + (−0.797 − 0.797i)5-s + 1.24i·7-s − 1.36i·9-s + (1.11 + 1.11i)11-s + (0.239 − 0.239i)13-s + 1.73·15-s − 1.65·17-s + (−0.589 + 0.589i)19-s + (−1.35 − 1.35i)21-s − 0.635i·23-s + 0.273i·25-s + (0.391 + 0.391i)27-s + (0.692 − 0.692i)29-s − 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.923 - 0.382i$
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: \((2,\ 128,\ (\ :11/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.6447824326\)
\(L(\frac12)\) \(\approx\) \(0.6447824326\)
\(L(\frac{13}{2})\) 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 \( 1 + (457. - 457. i)T - 1.77e5iT^{2} \)
5 \( 1 + (5.57e3 + 5.57e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 5.54e4iT - 1.97e9T^{2} \)
11 \( 1 + (-5.93e5 - 5.93e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-3.20e5 + 3.20e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 9.66e6T + 3.42e13T^{2} \)
19 \( 1 + (6.35e6 - 6.35e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 1.96e7iT - 9.52e14T^{2} \)
29 \( 1 + (-7.64e7 + 7.64e7i)T - 1.22e16iT^{2} \)
31 \( 1 + 1.66e8T + 2.54e16T^{2} \)
37 \( 1 + (2.74e7 + 2.74e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 6.15e8iT - 5.50e17T^{2} \)
43 \( 1 + (8.26e8 + 8.26e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.20e9T + 2.47e18T^{2} \)
53 \( 1 + (6.02e8 + 6.02e8i)T + 9.26e18iT^{2} \)
59 \( 1 + (3.88e9 + 3.88e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-2.43e9 + 2.43e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (8.86e9 - 8.86e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 2.93e10iT - 2.31e20T^{2} \)
73 \( 1 - 7.45e9iT - 3.13e20T^{2} \)
79 \( 1 + 2.31e10T + 7.47e20T^{2} \)
83 \( 1 + (4.40e9 - 4.40e9i)T - 1.28e21iT^{2} \)
89 \( 1 + 4.82e10iT - 2.77e21T^{2} \)
97 \( 1 - 9.19e10T + 7.15e21T^{2} \)
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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

−11.46124649926726691089765177604, −10.32783602465791708034571394349, −9.169048521034884047276323825508, −8.528497224053607530443997889141, −6.71253192696378152903349134447, −5.64214018735663289861156167083, −4.55100726545688717461094258552, −4.04473630712870030719227685118, −2.02816667517514966047778845915, −0.31811139050434668335293758753, 0.54502813077635856235812936920, 1.54059590730915893899756818106, 3.37265018002806889486618963670, 4.47028594894253337411419252252, 6.24092185852901609441053262477, 6.78964815588166012083588636141, 7.52340596510120570084073914689, 8.904686902436631950916853819778, 10.88640601092085482171647922781, 11.05628713039814878981661257950

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