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: $\chi_{128} (79, \cdot )$
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

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

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