Properties

Degree $2$
Conductor $128$
Sign $-0.999 + 0.0447i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.49 − 1.03i)3-s + (−0.452 + 0.187i)5-s + (−0.429 − 0.429i)7-s + (−1.19 − 1.19i)9-s + (−17.3 + 7.18i)11-s + (−19.9 − 8.26i)13-s + 1.32·15-s + 13.5i·17-s + (3.45 − 8.34i)19-s + (0.628 + 1.51i)21-s + (16.8 − 16.8i)23-s + (−17.5 + 17.5i)25-s + (11.0 + 26.7i)27-s + (13.8 − 33.4i)29-s − 24.5i·31-s + ⋯
L(s)  = 1  + (−0.832 − 0.344i)3-s + (−0.0904 + 0.0374i)5-s + (−0.0614 − 0.0614i)7-s + (−0.133 − 0.133i)9-s + (−1.57 + 0.652i)11-s + (−1.53 − 0.635i)13-s + 0.0882·15-s + 0.799i·17-s + (0.182 − 0.439i)19-s + (0.0299 + 0.0722i)21-s + (0.734 − 0.734i)23-s + (−0.700 + 0.700i)25-s + (0.409 + 0.989i)27-s + (0.477 − 1.15i)29-s − 0.792i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.999 + 0.0447i$
Motivic weight: \(2\)
Character: $\chi_{128} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.999 + 0.0447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00359774 - 0.160870i\)
\(L(\frac12)\) \(\approx\) \(0.00359774 - 0.160870i\)
\(L(2)\) 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 \( 1 + (2.49 + 1.03i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.452 - 0.187i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (0.429 + 0.429i)T + 49iT^{2} \)
11 \( 1 + (17.3 - 7.18i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (19.9 + 8.26i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 13.5iT - 289T^{2} \)
19 \( 1 + (-3.45 + 8.34i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \)
29 \( 1 + (-13.8 + 33.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 24.5iT - 961T^{2} \)
37 \( 1 + (-9.89 + 4.09i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-14.4 - 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (17.8 - 7.39i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 43.6T + 2.20e3T^{2} \)
53 \( 1 + (-28.0 - 67.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (1.70 + 4.10i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-3.53 + 8.53i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-0.300 - 0.124i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-29.0 - 29.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (68.2 + 68.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 67.7T + 6.24e3T^{2} \)
83 \( 1 + (-16.4 + 39.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (45.3 - 45.3i)T - 7.92e3iT^{2} \)
97 \( 1 + 119.T + 9.40e3T^{2} \)
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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

−12.61798857608085107851131035092, −11.66376045573694275917648566383, −10.57721828065347515734719526999, −9.705120440238937729111395109532, −8.038426787321906586079202049429, −7.13096838175263320617317417607, −5.77090727935254953552984273159, −4.78238692896011023963437579260, −2.64474418849821778702647610323, −0.11265602983737843590851207695, 2.77003796336917359576755374787, 4.84616399358111838630653486358, 5.50733914727349725063545309945, 7.07560051741842189015874525516, 8.222235215136665662081297343779, 9.686227272384058492412797308892, 10.58189014790870869626435847882, 11.52253618965087959498769431354, 12.38723901373277148182040506358, 13.58865525042637566752797572594

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