Properties

Degree $2$
Conductor $128$
Sign $0.962 + 0.270i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 − 0.568i)3-s + (2.28 + 0.948i)5-s + (6.37 − 6.37i)7-s + (−4.80 + 4.80i)9-s + (1.79 + 0.744i)11-s + (16.7 − 6.91i)13-s + 3.68·15-s − 6.19i·17-s + (8.50 + 20.5i)19-s + (5.12 − 12.3i)21-s + (−23.6 − 23.6i)23-s + (−13.3 − 13.3i)25-s + (−8.98 + 21.6i)27-s + (14.5 + 35.1i)29-s + 14.1i·31-s + ⋯
L(s)  = 1  + (0.457 − 0.189i)3-s + (0.457 + 0.189i)5-s + (0.911 − 0.911i)7-s + (−0.533 + 0.533i)9-s + (0.163 + 0.0676i)11-s + (1.28 − 0.532i)13-s + 0.245·15-s − 0.364i·17-s + (0.447 + 1.08i)19-s + (0.244 − 0.589i)21-s + (−1.02 − 1.02i)23-s + (−0.533 − 0.533i)25-s + (−0.332 + 0.802i)27-s + (0.502 + 1.21i)29-s + 0.456i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.962 + 0.270i$
Motivic weight: \(2\)
Character: $\chi_{128} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.962 + 0.270i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79382 - 0.247057i\)
\(L(\frac12)\) \(\approx\) \(1.79382 - 0.247057i\)
\(L(2)\) 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 \( 1 + (-1.37 + 0.568i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (-2.28 - 0.948i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (-6.37 + 6.37i)T - 49iT^{2} \)
11 \( 1 + (-1.79 - 0.744i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (-16.7 + 6.91i)T + (119. - 119. i)T^{2} \)
17 \( 1 + 6.19iT - 289T^{2} \)
19 \( 1 + (-8.50 - 20.5i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (23.6 + 23.6i)T + 529iT^{2} \)
29 \( 1 + (-14.5 - 35.1i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 14.1iT - 961T^{2} \)
37 \( 1 + (30.0 + 12.4i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (56.9 - 56.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (54.5 + 22.5i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 34.8T + 2.20e3T^{2} \)
53 \( 1 + (-3.92 + 9.48i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (9.41 - 22.7i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-3.00 - 7.25i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-55.9 + 23.1i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (6.27 - 6.27i)T - 5.04e3iT^{2} \)
73 \( 1 + (-66.4 + 66.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 75.8T + 6.24e3T^{2} \)
83 \( 1 + (-1.23 - 2.97i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-36.7 - 36.7i)T + 7.92e3iT^{2} \)
97 \( 1 - 90.0T + 9.40e3T^{2} \)
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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

−13.43909963980366777069135001510, −11.99877434413870766266564917808, −10.83582881152545811873057700520, −10.15452239998467214566956278953, −8.495435793312985712193602871282, −7.934298466347361330704115419408, −6.48647431988039642225924507441, −5.09987224828078336149453493395, −3.49297128134474606356853196318, −1.66089339828569808951378051597, 1.90617077908109698050909520928, 3.65084416137605233741657250502, 5.31135857980785873573836706298, 6.34475760136716914408754834507, 8.160812863017474858996369494201, 8.854622843024583018147726038499, 9.767375993737755637216620554782, 11.40768695798851242055084193208, 11.82505573050704145177993800187, 13.47002122975604834593607923312

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