Properties

Degree $2$
Conductor $128$
Sign $-0.300 - 0.953i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.57 − 3.13i)3-s + (−8.03 − 19.4i)5-s + (1.85 − 1.85i)7-s + (28.3 + 28.3i)9-s + (−8.63 + 3.57i)11-s + (−11.5 + 27.9i)13-s + 172. i·15-s − 7.99i·17-s + (−5.76 + 13.9i)19-s + (−19.8 + 8.20i)21-s + (−60.2 − 60.2i)23-s + (−223. + 223. i)25-s + (−41.2 − 99.4i)27-s + (167. + 69.3i)29-s + 225.·31-s + ⋯
L(s)  = 1  + (−1.45 − 0.603i)3-s + (−0.718 − 1.73i)5-s + (0.0999 − 0.0999i)7-s + (1.05 + 1.05i)9-s + (−0.236 + 0.0979i)11-s + (−0.247 + 0.597i)13-s + 2.96i·15-s − 0.114i·17-s + (−0.0695 + 0.167i)19-s + (−0.205 + 0.0853i)21-s + (−0.546 − 0.546i)23-s + (−1.78 + 1.78i)25-s + (−0.293 − 0.709i)27-s + (1.07 + 0.444i)29-s + 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.300 - 0.953i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.300 - 0.953i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0421230 + 0.0574658i\)
\(L(\frac12)\) \(\approx\) \(0.0421230 + 0.0574658i\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.57 + 3.13i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (8.03 + 19.4i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-1.85 + 1.85i)T - 343iT^{2} \)
11 \( 1 + (8.63 - 3.57i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (11.5 - 27.9i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 7.99iT - 4.91e3T^{2} \)
19 \( 1 + (5.76 - 13.9i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (60.2 + 60.2i)T + 1.21e4iT^{2} \)
29 \( 1 + (-167. - 69.3i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 225.T + 2.97e4T^{2} \)
37 \( 1 + (-0.431 - 1.04i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (275. + 275. i)T + 6.89e4iT^{2} \)
43 \( 1 + (257. - 106. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 51.3iT - 1.03e5T^{2} \)
53 \( 1 + (2.98 - 1.23i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (101. + 244. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (270. + 111. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (778. + 322. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (484. - 484. i)T - 3.57e5iT^{2} \)
73 \( 1 + (212. + 212. i)T + 3.89e5iT^{2} \)
79 \( 1 - 593. iT - 4.93e5T^{2} \)
83 \( 1 + (320. - 773. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-435. + 435. i)T - 7.04e5iT^{2} \)
97 \( 1 + 570.T + 9.12e5T^{2} \)
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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

−12.12026430480763054698712613713, −11.58329132284101015509885882139, −10.24712692637165269839294328432, −8.788160060435697840656634165390, −7.75733761532722814106111942317, −6.46464671506500618437363224304, −5.16630055967086416875574008184, −4.44573526349238147850573117155, −1.30611018587938653550858511905, −0.04837821624346476269289734211, 3.07446109384852445646649008735, 4.52130880434492146039730264164, 5.93952527339901391327614437338, 6.80443025531803635271355293177, 8.011663990340480324039428070290, 10.18057574479742561278366181184, 10.41299525362709484801309044274, 11.59963222331296672536130769252, 11.90715430684543874254685960903, 13.62047229785038842121974785663

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