Properties

Degree $2$
Conductor $128$
Sign $0.618 - 0.786i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.56 + 2.30i)3-s + (6.28 + 15.1i)5-s + (16.6 − 16.6i)7-s + (6.60 + 6.60i)9-s + (−3.11 + 1.28i)11-s + (−28.8 + 69.7i)13-s + 98.9i·15-s − 66.2i·17-s + (12.5 − 30.2i)19-s + (131. − 54.3i)21-s + (63.7 + 63.7i)23-s + (−102. + 102. i)25-s + (−40.7 − 98.3i)27-s + (190. + 79.0i)29-s − 123.·31-s + ⋯
L(s)  = 1  + (1.07 + 0.443i)3-s + (0.562 + 1.35i)5-s + (0.899 − 0.899i)7-s + (0.244 + 0.244i)9-s + (−0.0853 + 0.0353i)11-s + (−0.616 + 1.48i)13-s + 1.70i·15-s − 0.944i·17-s + (0.151 − 0.365i)19-s + (1.36 − 0.564i)21-s + (0.577 + 0.577i)23-s + (−0.818 + 0.818i)25-s + (−0.290 − 0.701i)27-s + (1.22 + 0.506i)29-s − 0.717·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.618 - 0.786i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.618 - 0.786i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.29466 + 1.11484i\)
\(L(\frac12)\) \(\approx\) \(2.29466 + 1.11484i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.56 - 2.30i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-6.28 - 15.1i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-16.6 + 16.6i)T - 343iT^{2} \)
11 \( 1 + (3.11 - 1.28i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (28.8 - 69.7i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 66.2iT - 4.91e3T^{2} \)
19 \( 1 + (-12.5 + 30.2i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-63.7 - 63.7i)T + 1.21e4iT^{2} \)
29 \( 1 + (-190. - 79.0i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 123.T + 2.97e4T^{2} \)
37 \( 1 + (46.0 + 111. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (100. + 100. i)T + 6.89e4iT^{2} \)
43 \( 1 + (27.5 - 11.4i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 394. iT - 1.03e5T^{2} \)
53 \( 1 + (-135. + 56.0i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (297. + 717. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (548. + 227. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (163. + 67.6i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (194. - 194. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-547. - 547. i)T + 3.89e5iT^{2} \)
79 \( 1 + 715. iT - 4.93e5T^{2} \)
83 \( 1 + (54.6 - 131. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-220. + 220. i)T - 7.04e5iT^{2} \)
97 \( 1 - 1.36e3T + 9.12e5T^{2} \)
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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

−13.65121415868600729157934674545, −11.71719816019690992390341320446, −10.82009392697927014589671309867, −9.836879785515965498782728498526, −8.958664689476609142774926816536, −7.49526129695723536271583944801, −6.77271856492342353415196865520, −4.78553638461705478455651034300, −3.39433716575468759670245980199, −2.14280278900131243648995711450, 1.44216417299400771096273970935, 2.73159970132421638010734824911, 4.83574632672533679814133032223, 5.76597392425619441733929809303, 7.85804337160910060046290951712, 8.405202192246651723288052965915, 9.168941639465349511822690085477, 10.49288092823060455795094880571, 12.14417074148669826432739858327, 12.78065158743296577069505335996

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