Properties

Degree $2$
Conductor $128$
Sign $-0.951 + 0.307i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.10 − 5.07i)3-s + (1.74 − 4.21i)5-s + (0.392 − 0.392i)7-s + (−14.9 + 14.9i)9-s + (−2.90 + 7.02i)11-s + (−4.50 − 10.8i)13-s − 25.0·15-s − 10.5i·17-s + (1.88 − 0.781i)19-s + (−2.81 − 1.16i)21-s + (0.445 + 0.445i)23-s + (2.94 + 2.94i)25-s + (61.7 + 25.5i)27-s + (0.741 − 0.307i)29-s − 47.6i·31-s + ⋯
L(s)  = 1  + (−0.700 − 1.69i)3-s + (0.349 − 0.843i)5-s + (0.0560 − 0.0560i)7-s + (−1.66 + 1.66i)9-s + (−0.264 + 0.638i)11-s + (−0.346 − 0.836i)13-s − 1.67·15-s − 0.620i·17-s + (0.0993 − 0.0411i)19-s + (−0.134 − 0.0555i)21-s + (0.0193 + 0.0193i)23-s + (0.117 + 0.117i)25-s + (2.28 + 0.947i)27-s + (0.0255 − 0.0105i)29-s − 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.951 + 0.307i$
Motivic weight: \(2\)
Character: $\chi_{128} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.951 + 0.307i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.146928 - 0.933114i\)
\(L(\frac12)\) \(\approx\) \(0.146928 - 0.933114i\)
\(L(2)\) 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 \( 1 + (2.10 + 5.07i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-1.74 + 4.21i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-0.392 + 0.392i)T - 49iT^{2} \)
11 \( 1 + (2.90 - 7.02i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (4.50 + 10.8i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 10.5iT - 289T^{2} \)
19 \( 1 + (-1.88 + 0.781i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-0.445 - 0.445i)T + 529iT^{2} \)
29 \( 1 + (-0.741 + 0.307i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 47.6iT - 961T^{2} \)
37 \( 1 + (-14.5 + 35.0i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (11.3 - 11.3i)T - 1.68e3iT^{2} \)
43 \( 1 + (-14.6 + 35.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 80.5T + 2.20e3T^{2} \)
53 \( 1 + (66.6 + 27.5i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (65.0 + 26.9i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-87.4 + 36.2i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-7.12 - 17.1i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-14.8 + 14.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (-18.6 + 18.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 36.2T + 6.24e3T^{2} \)
83 \( 1 + (27.0 - 11.2i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (56.4 + 56.4i)T + 7.92e3iT^{2} \)
97 \( 1 - 158.T + 9.40e3T^{2} \)
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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

−12.73385155411433182826806246153, −11.96203054774400265366238752059, −10.86098152631795895255061142563, −9.382117834958963318476767952299, −7.979207671417954223943696884197, −7.25344439728727104552021715638, −5.93210525300873223842617461167, −5.02032778318123774003871169503, −2.25698061221226958139956072131, −0.70776378552767054851160259511, 3.09645967259118585887825381079, 4.42123155581122845822645705997, 5.63020169171113344650566869315, 6.66939811097103589437223129842, 8.615218059732384358307622283329, 9.693512101360624177733354969335, 10.53564235238982637648230643359, 11.13280552674211530666055798444, 12.19304324866609904857103561772, 13.94098174620510621955573472546

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