Properties

Degree $2$
Conductor $128$
Sign $-0.208 + 0.978i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.66 + 1.10i)3-s + (−5.50 − 13.2i)5-s + (6.48 − 6.48i)7-s + (−13.2 − 13.2i)9-s + (−49.3 + 20.4i)11-s + (21.4 − 51.7i)13-s − 41.4i·15-s − 3.73i·17-s + (36.7 − 88.6i)19-s + (24.4 − 10.1i)21-s + (45.4 + 45.4i)23-s + (−58.0 + 58.0i)25-s + (−50.3 − 121. i)27-s + (51.9 + 21.5i)29-s + 73.5·31-s + ⋯
L(s)  = 1  + (0.512 + 0.212i)3-s + (−0.492 − 1.18i)5-s + (0.349 − 0.349i)7-s + (−0.489 − 0.489i)9-s + (−1.35 + 0.560i)11-s + (0.457 − 1.10i)13-s − 0.714i·15-s − 0.0533i·17-s + (0.443 − 1.07i)19-s + (0.253 − 0.105i)21-s + (0.412 + 0.412i)23-s + (−0.464 + 0.464i)25-s + (−0.359 − 0.867i)27-s + (0.332 + 0.137i)29-s + 0.425·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.208 + 0.978i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.208 + 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.865643 - 1.06919i\)
\(L(\frac12)\) \(\approx\) \(0.865643 - 1.06919i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.66 - 1.10i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (5.50 + 13.2i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-6.48 + 6.48i)T - 343iT^{2} \)
11 \( 1 + (49.3 - 20.4i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-21.4 + 51.7i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 3.73iT - 4.91e3T^{2} \)
19 \( 1 + (-36.7 + 88.6i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-45.4 - 45.4i)T + 1.21e4iT^{2} \)
29 \( 1 + (-51.9 - 21.5i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 73.5T + 2.97e4T^{2} \)
37 \( 1 + (165. + 399. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-334. - 334. i)T + 6.89e4iT^{2} \)
43 \( 1 + (328. - 136. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 185. iT - 1.03e5T^{2} \)
53 \( 1 + (-412. + 171. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-214. - 518. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-85.1 - 35.2i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-252. - 104. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (430. - 430. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-41.8 - 41.8i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.21e3iT - 4.93e5T^{2} \)
83 \( 1 + (-290. + 702. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-365. + 365. i)T - 7.04e5iT^{2} \)
97 \( 1 - 508.T + 9.12e5T^{2} \)
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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

−12.77907172677324161520880791149, −11.61109050376782705412986569089, −10.46305667577933194502354953506, −9.216121544123117964837922191831, −8.323364566827366309458808498952, −7.51012323802255056080625890426, −5.52221749682003105354276019555, −4.50321165334236459703558934187, −2.95039680324672352761605531410, −0.66340199389947310529863983884, 2.31930289595695739450919008457, 3.45580296850073357969648209200, 5.32064726349451282242628838932, 6.74739257096345812951636703375, 7.908849771835775229723545311672, 8.604329955342657596415733771183, 10.27082913640687818144306256595, 11.08497064551719773416957621174, 11.94959865330546382693401997910, 13.47639196480117910987861600854

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