Properties

Degree $2$
Conductor $128$
Sign $-0.405 + 0.914i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.31 − 3.18i)3-s + (−0.659 − 1.59i)5-s + (−9.54 − 9.54i)7-s + (−2.03 − 2.03i)9-s + (3.96 + 9.57i)11-s + (1.91 − 4.63i)13-s − 5.93·15-s − 15.3i·17-s + (−0.827 − 0.342i)19-s + (−42.9 + 17.8i)21-s + (12.9 − 12.9i)23-s + (15.5 − 15.5i)25-s + (19.5 − 8.07i)27-s + (23.7 + 9.85i)29-s + 25.1i·31-s + ⋯
L(s)  = 1  + (0.439 − 1.06i)3-s + (−0.131 − 0.318i)5-s + (−1.36 − 1.36i)7-s + (−0.225 − 0.225i)9-s + (0.360 + 0.870i)11-s + (0.147 − 0.356i)13-s − 0.395·15-s − 0.900i·17-s + (−0.0435 − 0.0180i)19-s + (−2.04 + 0.847i)21-s + (0.561 − 0.561i)23-s + (0.623 − 0.623i)25-s + (0.722 − 0.299i)27-s + (0.820 + 0.339i)29-s + 0.811i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.405 + 0.914i$
Motivic weight: \(2\)
Character: $\chi_{128} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.405 + 0.914i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.725339 - 1.11511i\)
\(L(\frac12)\) \(\approx\) \(0.725339 - 1.11511i\)
\(L(2)\) 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 \( 1 + (-1.31 + 3.18i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (0.659 + 1.59i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (9.54 + 9.54i)T + 49iT^{2} \)
11 \( 1 + (-3.96 - 9.57i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-1.91 + 4.63i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 15.3iT - 289T^{2} \)
19 \( 1 + (0.827 + 0.342i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-12.9 + 12.9i)T - 529iT^{2} \)
29 \( 1 + (-23.7 - 9.85i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 25.1iT - 961T^{2} \)
37 \( 1 + (-13.6 - 32.8i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (32.9 + 32.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-17.9 - 43.3i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 20.1T + 2.20e3T^{2} \)
53 \( 1 + (-35.0 + 14.5i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-60.6 + 25.1i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (27.9 + 11.5i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (1.13 - 2.73i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-45.6 - 45.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (29.1 + 29.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 3.27T + 6.24e3T^{2} \)
83 \( 1 + (56.7 + 23.5i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (44.5 - 44.5i)T - 7.92e3iT^{2} \)
97 \( 1 + 106.T + 9.40e3T^{2} \)
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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

−12.87454336530299983634796445284, −12.20692673291195367424822341850, −10.55643974900128061020925239365, −9.667223450595342410059370273803, −8.335884069439091399177040225853, −7.04759140614157899441845207482, −6.73504693491345690691247878601, −4.55725229093739476605781812022, −2.97823209364575878504002127509, −0.925553276069034001248060961164, 2.90582805336735324486445327000, 3.84406768837603127247405278968, 5.63310755337178767383526287127, 6.65935768053427438930244613397, 8.594731967834538929869556775138, 9.218021286188353442950559432760, 10.10304650911722896697481343985, 11.28502329411934234919821819788, 12.41122663430762276739177589043, 13.42808670539641734838618704336

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