Properties

Degree $2$
Conductor $128$
Sign $0.975 + 0.220i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.06 + 2.51i)3-s + (−2.91 − 7.02i)5-s + (13.3 − 13.3i)7-s + (11.3 + 11.3i)9-s + (49.6 − 20.5i)11-s + (8.74 − 21.1i)13-s − 49.9i·15-s + 77.7i·17-s + (−53.3 + 128. i)19-s + (114. − 47.5i)21-s + (35.5 + 35.5i)23-s + (47.4 − 47.4i)25-s + (−27.3 − 66.0i)27-s + (−245. − 101. i)29-s + 202.·31-s + ⋯
L(s)  = 1  + (1.16 + 0.483i)3-s + (−0.260 − 0.628i)5-s + (0.722 − 0.722i)7-s + (0.421 + 0.421i)9-s + (1.36 − 0.563i)11-s + (0.186 − 0.450i)13-s − 0.859i·15-s + 1.10i·17-s + (−0.643 + 1.55i)19-s + (1.19 − 0.494i)21-s + (0.321 + 0.321i)23-s + (0.379 − 0.379i)25-s + (−0.195 − 0.470i)27-s + (−1.57 − 0.650i)29-s + 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.975 + 0.220i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.44767 - 0.273525i\)
\(L(\frac12)\) \(\approx\) \(2.44767 - 0.273525i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.06 - 2.51i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (2.91 + 7.02i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-13.3 + 13.3i)T - 343iT^{2} \)
11 \( 1 + (-49.6 + 20.5i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-8.74 + 21.1i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 77.7iT - 4.91e3T^{2} \)
19 \( 1 + (53.3 - 128. i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-35.5 - 35.5i)T + 1.21e4iT^{2} \)
29 \( 1 + (245. + 101. i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 202.T + 2.97e4T^{2} \)
37 \( 1 + (-36.3 - 87.6i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (36.8 + 36.8i)T + 6.89e4iT^{2} \)
43 \( 1 + (185. - 76.8i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 82.9iT - 1.03e5T^{2} \)
53 \( 1 + (534. - 221. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (75.7 + 182. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (472. + 195. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-102. - 42.5i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (520. - 520. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-244. - 244. i)T + 3.89e5iT^{2} \)
79 \( 1 - 774. iT - 4.93e5T^{2} \)
83 \( 1 + (-23.9 + 57.7i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (351. - 351. i)T - 7.04e5iT^{2} \)
97 \( 1 - 1.30e3T + 9.12e5T^{2} \)
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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

−13.00023093426471128043966147434, −11.81052642267436957517751946660, −10.64137214325514763516962579625, −9.510925159648570006066310834784, −8.425180100998218713159713694318, −7.978071963036125074773467156435, −6.15256370808150071241082092485, −4.31431968295382397860158560280, −3.59332476483093028158685445912, −1.44066552882439602836749417894, 1.87031967379275742566207845755, 3.09148379133464573276555354607, 4.71779724871614446441042095707, 6.67806905884174346022904161890, 7.47301951240700067033775802646, 8.822814021539199389424767183364, 9.256276095916455948868153542592, 11.10285355700958631726107209016, 11.79745712943668889573110086388, 13.08258921835907997862327447205

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