Properties

Label 2-2e7-16.13-c11-0-32
Degree $2$
Conductor $128$
Sign $-0.499 + 0.866i$
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  + (−137. − 137. i)3-s + (−3.89e3 + 3.89e3i)5-s − 3.79e4i·7-s − 1.39e5i·9-s + (6.81e4 − 6.81e4i)11-s + (1.09e5 + 1.09e5i)13-s + 1.06e6·15-s + 8.85e6·17-s + (7.12e5 + 7.12e5i)19-s + (−5.19e6 + 5.19e6i)21-s − 2.69e7i·23-s + 1.84e7i·25-s + (−4.34e7 + 4.34e7i)27-s + (1.10e8 + 1.10e8i)29-s − 1.15e8·31-s + ⋯
L(s)  = 1  + (−0.325 − 0.325i)3-s + (−0.558 + 0.558i)5-s − 0.853i·7-s − 0.787i·9-s + (0.127 − 0.127i)11-s + (0.0820 + 0.0820i)13-s + 0.363·15-s + 1.51·17-s + (0.0660 + 0.0660i)19-s + (−0.277 + 0.277i)21-s − 0.873i·23-s + 0.377i·25-s + (−0.582 + 0.582i)27-s + (0.999 + 0.999i)29-s − 0.725·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.499 + 0.866i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ -0.499 + 0.866i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.312986485\)
\(L(\frac12)\) \(\approx\) \(1.312986485\)
\(L(\frac{13}{2})\) 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 \( 1 + (137. + 137. i)T + 1.77e5iT^{2} \)
5 \( 1 + (3.89e3 - 3.89e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 3.79e4iT - 1.97e9T^{2} \)
11 \( 1 + (-6.81e4 + 6.81e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (-1.09e5 - 1.09e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 8.85e6T + 3.42e13T^{2} \)
19 \( 1 + (-7.12e5 - 7.12e5i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.69e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.10e8 - 1.10e8i)T + 1.22e16iT^{2} \)
31 \( 1 + 1.15e8T + 2.54e16T^{2} \)
37 \( 1 + (3.15e8 - 3.15e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 8.66e8iT - 5.50e17T^{2} \)
43 \( 1 + (-3.13e8 + 3.13e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.93e9T + 2.47e18T^{2} \)
53 \( 1 + (8.24e8 - 8.24e8i)T - 9.26e18iT^{2} \)
59 \( 1 + (-7.47e9 + 7.47e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-6.44e8 - 6.44e8i)T + 4.35e19iT^{2} \)
67 \( 1 + (5.39e9 + 5.39e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 2.90e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.12e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.03e7T + 7.47e20T^{2} \)
83 \( 1 + (2.10e10 + 2.10e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 7.01e10iT - 2.77e21T^{2} \)
97 \( 1 + 3.85e10T + 7.15e21T^{2} \)
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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

−10.89180861928765648307879175793, −10.05973422439517320926062286711, −8.698908734063190770170565632923, −7.39601008810688380103714442389, −6.80371439322589311466996081743, −5.52593623991437491842385706037, −3.96229113235906409177790548646, −3.15821136209258676670972038469, −1.29590396063154078753735455756, −0.37200276052773935253892040212, 1.04252770905877169663648613436, 2.48157555282153582238430389400, 3.92188819242368757349776192391, 5.08801063849975551967129375706, 5.85802175566301471496573281036, 7.54742609633051023905918472499, 8.365834631420284702081703180830, 9.516585653371336343684233723590, 10.54338581756925774755732852187, 11.75376699776924763260178827990

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