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: $\chi_{128} (49, \cdot )$
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

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

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