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: $\chi_{128} (49, \cdot )$
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

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

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