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Properties

Label 2-2e7-16.5-c11-0-28
Degree $2$
Conductor $128$
Sign $0.988 - 0.151i$
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  + (−221. + 221. i)3-s + (7.91e3 + 7.91e3i)5-s − 7.68e4i·7-s + 7.90e4i·9-s + (1.97e5 + 1.97e5i)11-s + (4.68e4 − 4.68e4i)13-s − 3.50e6·15-s + 9.74e6·17-s + (−2.85e6 + 2.85e6i)19-s + (1.70e7 + 1.70e7i)21-s − 3.12e7i·23-s + 7.65e7i·25-s + (−5.67e7 − 5.67e7i)27-s + (1.26e8 − 1.26e8i)29-s + 3.46e7·31-s + ⋯
L(s)  = 1  + (−0.526 + 0.526i)3-s + (1.13 + 1.13i)5-s − 1.72i·7-s + 0.446i·9-s + (0.370 + 0.370i)11-s + (0.0350 − 0.0350i)13-s − 1.19·15-s + 1.66·17-s + (−0.264 + 0.264i)19-s + (0.909 + 0.909i)21-s − 1.01i·23-s + 1.56i·25-s + (−0.761 − 0.761i)27-s + (1.14 − 1.14i)29-s + 0.217·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.988 - 0.151i$
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: \((2,\ 128,\ (\ :11/2),\ 0.988 - 0.151i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.477683402\)
\(L(\frac12)\) \(\approx\) \(2.477683402\)
\(L(\frac{13}{2})\) 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 \( 1 + (221. - 221. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-7.91e3 - 7.91e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 7.68e4iT - 1.97e9T^{2} \)
11 \( 1 + (-1.97e5 - 1.97e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-4.68e4 + 4.68e4i)T - 1.79e12iT^{2} \)
17 \( 1 - 9.74e6T + 3.42e13T^{2} \)
19 \( 1 + (2.85e6 - 2.85e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 3.12e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.26e8 + 1.26e8i)T - 1.22e16iT^{2} \)
31 \( 1 - 3.46e7T + 2.54e16T^{2} \)
37 \( 1 + (2.25e8 + 2.25e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 4.55e8iT - 5.50e17T^{2} \)
43 \( 1 + (7.03e8 + 7.03e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 6.32e8T + 2.47e18T^{2} \)
53 \( 1 + (-1.37e9 - 1.37e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (1.10e9 + 1.10e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (3.14e9 - 3.14e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-3.21e9 + 3.21e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.72e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.53e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.44e10T + 7.47e20T^{2} \)
83 \( 1 + (-1.61e10 + 1.61e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 1.60e10iT - 2.77e21T^{2} \)
97 \( 1 - 7.16e9T + 7.15e21T^{2} \)
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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

−10.63378694219987423742062952753, −10.45156293773220603437601545244, −9.749747874025722534258797180560, −7.82401161718154338021111426920, −6.84998474107979578188534634889, −5.90756549940572470807871729221, −4.62022015629973439208332000081, −3.47994800931527873651365528070, −2.05476543213346898869607865442, −0.70023040995493751432773391041, 0.979929187910998783040093234622, 1.66712632654371953557039105247, 3.09241013210452102931659059583, 5.15396725613945965296019638000, 5.67278002985367219550381039206, 6.51549775909651744001741401305, 8.315587504719037409884500329104, 9.122177350448828345245038270333, 9.858806194712714707067572600581, 11.62972965426240650340090154095

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