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: $\chi_{128} (15, \cdot )$
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

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

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