Properties

Degree $2$
Conductor $128$
Sign $0.997 - 0.0673i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.70 + 1.53i)3-s + (7.20 − 2.98i)5-s + (−4.26 − 4.26i)7-s + (4.99 + 4.99i)9-s + (−6.19 + 2.56i)11-s + (−8.05 − 3.33i)13-s + 31.2·15-s + 24.5i·17-s + (−4.96 + 11.9i)19-s + (−9.24 − 22.3i)21-s + (9.72 − 9.72i)23-s + (25.2 − 25.2i)25-s + (−2.97 − 7.17i)27-s + (−5.86 + 14.1i)29-s + 17.5i·31-s + ⋯
L(s)  = 1  + (1.23 + 0.511i)3-s + (1.44 − 0.596i)5-s + (−0.608 − 0.608i)7-s + (0.554 + 0.554i)9-s + (−0.563 + 0.233i)11-s + (−0.619 − 0.256i)13-s + 2.08·15-s + 1.44i·17-s + (−0.261 + 0.630i)19-s + (−0.440 − 1.06i)21-s + (0.422 − 0.422i)23-s + (1.01 − 1.01i)25-s + (−0.110 − 0.265i)27-s + (−0.202 + 0.488i)29-s + 0.565i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.997 - 0.0673i$
Motivic weight: \(2\)
Character: $\chi_{128} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.997 - 0.0673i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.19517 + 0.0740419i\)
\(L(\frac12)\) \(\approx\) \(2.19517 + 0.0740419i\)
\(L(2)\) 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 \( 1 + (-3.70 - 1.53i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-7.20 + 2.98i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (4.26 + 4.26i)T + 49iT^{2} \)
11 \( 1 + (6.19 - 2.56i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (8.05 + 3.33i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 24.5iT - 289T^{2} \)
19 \( 1 + (4.96 - 11.9i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-9.72 + 9.72i)T - 529iT^{2} \)
29 \( 1 + (5.86 - 14.1i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 17.5iT - 961T^{2} \)
37 \( 1 + (36.0 - 14.9i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-10.9 - 10.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-22.4 + 9.27i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 27.0T + 2.20e3T^{2} \)
53 \( 1 + (34.0 + 82.1i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (27.8 + 67.2i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (6.37 - 15.3i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-99.2 - 41.0i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-2.55 - 2.55i)T + 5.04e3iT^{2} \)
73 \( 1 + (-30.7 - 30.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 90.6T + 6.24e3T^{2} \)
83 \( 1 + (-39.3 + 94.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-109. + 109. i)T - 7.92e3iT^{2} \)
97 \( 1 - 63.7T + 9.40e3T^{2} \)
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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

−13.13497985246314597726296178898, −12.67364105647083219871104737450, −10.40765809865086821492583636839, −9.971859608304203654485092135568, −9.015678383737987501337863523377, −8.066993817049354322742023113864, −6.46327199316125872919574640723, −5.06446553576036404111858790758, −3.51604107213117156114464625924, −2.03346270234726910274027633146, 2.29293047667956964396802553026, 2.90568650194143099807279198115, 5.35591874173903899176829657509, 6.62400455106592350408839000285, 7.66029916436650120534532571355, 9.250944290617086741824966258320, 9.451903661517283629476249085037, 10.87522076669686435516583933680, 12.42044566678166524039520662757, 13.51612615236990508957442795398

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