Properties

Degree $2$
Conductor $128$
Sign $0.979 + 0.202i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.97 − 2.89i)3-s + (1.57 + 3.79i)5-s + (−15.6 + 15.6i)7-s + (21.2 + 21.2i)9-s + (56.0 − 23.2i)11-s + (14.0 − 34.0i)13-s − 31.0i·15-s + 26.6i·17-s + (30.7 − 74.2i)19-s + (154. − 64.0i)21-s + (141. + 141. i)23-s + (76.4 − 76.4i)25-s + (−8.90 − 21.4i)27-s + (−11.9 − 4.97i)29-s + 128.·31-s + ⋯
L(s)  = 1  + (−1.34 − 0.556i)3-s + (0.140 + 0.339i)5-s + (−0.845 + 0.845i)7-s + (0.787 + 0.787i)9-s + (1.53 − 0.636i)11-s + (0.300 − 0.726i)13-s − 0.534i·15-s + 0.379i·17-s + (0.371 − 0.897i)19-s + (1.60 − 0.665i)21-s + (1.28 + 1.28i)23-s + (0.611 − 0.611i)25-s + (−0.0634 − 0.153i)27-s + (−0.0768 − 0.0318i)29-s + 0.744·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.979 + 0.202i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01361 - 0.103491i\)
\(L(\frac12)\) \(\approx\) \(1.01361 - 0.103491i\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.97 + 2.89i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.57 - 3.79i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (15.6 - 15.6i)T - 343iT^{2} \)
11 \( 1 + (-56.0 + 23.2i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-14.0 + 34.0i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 26.6iT - 4.91e3T^{2} \)
19 \( 1 + (-30.7 + 74.2i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-141. - 141. i)T + 1.21e4iT^{2} \)
29 \( 1 + (11.9 + 4.97i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 128.T + 2.97e4T^{2} \)
37 \( 1 + (-85.1 - 205. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (32.3 + 32.3i)T + 6.89e4iT^{2} \)
43 \( 1 + (314. - 130. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 184. iT - 1.03e5T^{2} \)
53 \( 1 + (-277. + 114. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (241. + 582. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-297. - 123. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-605. - 250. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (163. - 163. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-624. - 624. i)T + 3.89e5iT^{2} \)
79 \( 1 - 139. iT - 4.93e5T^{2} \)
83 \( 1 + (-226. + 545. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-231. + 231. i)T - 7.04e5iT^{2} \)
97 \( 1 + 594.T + 9.12e5T^{2} \)
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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

−12.68502652595215966118021865848, −11.70132370322960277291366901078, −11.16063677682438941330260763907, −9.790938024282760914807705999143, −8.663632779727648454769698654990, −6.83371127746428703822599506266, −6.29206350093990929054233456734, −5.27396047136393406317000494815, −3.21102818731696727946811140171, −0.954635136336723371357672091056, 0.951373998866872902390455289672, 3.85998000618974097399909870657, 4.86043462789372648212343641397, 6.32339078125495287215242701749, 6.97294258505993849054914643858, 9.048861807867331024756972894798, 9.896008041729980876985872143797, 10.85548999409712154160930820755, 11.83542052351896336569010604101, 12.62174306693446361117142243699

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