Properties

Degree $2$
Conductor $128$
Sign $0.730 - 0.682i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.35 − 1.80i)3-s + (−2.81 + 1.16i)5-s + (6.23 + 6.23i)7-s + (9.32 + 9.32i)9-s + (8.06 − 3.33i)11-s + (13.3 + 5.51i)13-s + 14.3·15-s + 4.56i·17-s + (−13.4 + 32.4i)19-s + (−15.8 − 38.3i)21-s + (6.75 − 6.75i)23-s + (−11.1 + 11.1i)25-s + (−7.53 − 18.1i)27-s + (−0.266 + 0.643i)29-s − 0.326i·31-s + ⋯
L(s)  = 1  + (−1.45 − 0.600i)3-s + (−0.563 + 0.233i)5-s + (0.890 + 0.890i)7-s + (1.03 + 1.03i)9-s + (0.732 − 0.303i)11-s + (1.02 + 0.424i)13-s + 0.957·15-s + 0.268i·17-s + (−0.707 + 1.70i)19-s + (−0.756 − 1.82i)21-s + (0.293 − 0.293i)23-s + (−0.444 + 0.444i)25-s + (−0.279 − 0.674i)27-s + (−0.00918 + 0.0221i)29-s − 0.0105i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.730 - 0.682i$
Motivic weight: \(2\)
Character: $\chi_{128} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.730 - 0.682i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.773027 + 0.304915i\)
\(L(\frac12)\) \(\approx\) \(0.773027 + 0.304915i\)
\(L(2)\) 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 \( 1 + (4.35 + 1.80i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (2.81 - 1.16i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-6.23 - 6.23i)T + 49iT^{2} \)
11 \( 1 + (-8.06 + 3.33i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-13.3 - 5.51i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 4.56iT - 289T^{2} \)
19 \( 1 + (13.4 - 32.4i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-6.75 + 6.75i)T - 529iT^{2} \)
29 \( 1 + (0.266 - 0.643i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 0.326iT - 961T^{2} \)
37 \( 1 + (-31.5 + 13.0i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-15.7 - 15.7i)T + 1.68e3iT^{2} \)
43 \( 1 + (4.83 - 2.00i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 49.7T + 2.20e3T^{2} \)
53 \( 1 + (-4.45 - 10.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (13.1 + 31.6i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (35.4 - 85.4i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-41.3 - 17.1i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (37.6 + 37.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (52.2 + 52.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 26.9T + 6.24e3T^{2} \)
83 \( 1 + (10.6 - 25.6i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (103. - 103. i)T - 7.92e3iT^{2} \)
97 \( 1 - 77.9T + 9.40e3T^{2} \)
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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

−12.84892767241598507119305027025, −11.91661401255535274666975019697, −11.44858992777269254887967377695, −10.61557096847915877104000336380, −8.820371581817671874745384259347, −7.73813404491080077774556889330, −6.33957623614254509725564592305, −5.67562141437160332711103054884, −4.11706811250453765820879660682, −1.51196201679853341688148725425, 0.78385676052385988870316334500, 4.10027521595925157963293911849, 4.80736251025598475738912557849, 6.20539479132503606926320840302, 7.39361602269803584946474010346, 8.804599789521064708230272487187, 10.22788243170872339092213975379, 11.22971491263147251902338484554, 11.45722414191111264555352904604, 12.76661550638573116142123950255

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