Properties

Degree $2$
Conductor $128$
Sign $0.241 - 0.970i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 + 0.655i)3-s + (4.18 + 1.73i)5-s + (−3.93 + 3.93i)7-s + (−4.29 + 4.29i)9-s + (14.2 + 5.89i)11-s + (0.454 − 0.188i)13-s − 7.76·15-s + 26.5i·17-s + (7.25 + 17.5i)19-s + (3.64 − 8.79i)21-s + (−0.775 − 0.775i)23-s + (−3.14 − 3.14i)25-s + (9.87 − 23.8i)27-s + (−17.9 − 43.4i)29-s − 39.6i·31-s + ⋯
L(s)  = 1  + (−0.527 + 0.218i)3-s + (0.837 + 0.346i)5-s + (−0.561 + 0.561i)7-s + (−0.476 + 0.476i)9-s + (1.29 + 0.536i)11-s + (0.0349 − 0.0144i)13-s − 0.517·15-s + 1.56i·17-s + (0.381 + 0.921i)19-s + (0.173 − 0.418i)21-s + (−0.0337 − 0.0337i)23-s + (−0.125 − 0.125i)25-s + (0.365 − 0.882i)27-s + (−0.620 − 1.49i)29-s − 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.241 - 0.970i$
Motivic weight: \(2\)
Character: $\chi_{128} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.241 - 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.957609 + 0.748702i\)
\(L(\frac12)\) \(\approx\) \(0.957609 + 0.748702i\)
\(L(2)\) 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 \( 1 + (1.58 - 0.655i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (-4.18 - 1.73i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (3.93 - 3.93i)T - 49iT^{2} \)
11 \( 1 + (-14.2 - 5.89i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (-0.454 + 0.188i)T + (119. - 119. i)T^{2} \)
17 \( 1 - 26.5iT - 289T^{2} \)
19 \( 1 + (-7.25 - 17.5i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (0.775 + 0.775i)T + 529iT^{2} \)
29 \( 1 + (17.9 + 43.4i)T + (-594. + 594. i)T^{2} \)
31 \( 1 + 39.6iT - 961T^{2} \)
37 \( 1 + (-36.4 - 15.1i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (-38.9 + 38.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (-14.2 - 5.91i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 62.1T + 2.20e3T^{2} \)
53 \( 1 + (-11.4 + 27.7i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-5.30 + 12.8i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-14.1 - 34.1i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (26.1 - 10.8i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (17.7 - 17.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (-12.8 + 12.8i)T - 5.32e3iT^{2} \)
79 \( 1 - 144.T + 6.24e3T^{2} \)
83 \( 1 + (-10.9 - 26.5i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (5.92 + 5.92i)T + 7.92e3iT^{2} \)
97 \( 1 - 66.9T + 9.40e3T^{2} \)
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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

−13.29344327600602739271513100547, −12.21576225862253609313114097343, −11.29380669646835572173744468713, −10.09939982526151448214130537538, −9.415908291705700233814141725899, −8.004735228180090839450072422624, −6.21250996086408186956755123153, −5.89163040118517158304796567791, −4.04937752268430795987073339433, −2.13479170008386890469948063222, 0.951323385517723561394941380470, 3.27787102914688834337076210433, 5.09454006427106012846960500245, 6.27371347809624572221800255568, 7.09877173181665578228851637900, 9.061475502534349855639204282395, 9.481123648938908263390514331370, 11.01581173497600485524821608566, 11.80575827625339428526762621076, 12.93203203054441231141448053537

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