Properties

Degree $2$
Conductor $128$
Sign $-0.977 - 0.209i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.21 − 7.75i)3-s + (−13.6 + 5.65i)5-s + (−9.07 + 9.07i)7-s + (−30.7 − 30.7i)9-s + (−16.7 − 40.3i)11-s + (−38.4 − 15.9i)13-s + 123. i·15-s + 92.5i·17-s + (16.6 + 6.90i)19-s + (41.2 + 99.5i)21-s + (−95.6 − 95.6i)23-s + (65.9 − 65.9i)25-s + (−127. + 52.8i)27-s + (19.3 − 46.7i)29-s + 38.1·31-s + ⋯
L(s)  = 1  + (0.618 − 1.49i)3-s + (−1.22 + 0.505i)5-s + (−0.490 + 0.490i)7-s + (−1.13 − 1.13i)9-s + (−0.458 − 1.10i)11-s + (−0.819 − 0.339i)13-s + 2.13i·15-s + 1.31i·17-s + (0.201 + 0.0834i)19-s + (0.428 + 1.03i)21-s + (−0.866 − 0.866i)23-s + (0.527 − 0.527i)25-s + (−0.909 + 0.376i)27-s + (0.123 − 0.299i)29-s + 0.221·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.977 - 0.209i$
Motivic weight: \(3\)
Character: $\chi_{128} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0598338 + 0.566231i\)
\(L(\frac12)\) \(\approx\) \(0.0598338 + 0.566231i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.21 + 7.75i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (13.6 - 5.65i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (9.07 - 9.07i)T - 343iT^{2} \)
11 \( 1 + (16.7 + 40.3i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (38.4 + 15.9i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 92.5iT - 4.91e3T^{2} \)
19 \( 1 + (-16.6 - 6.90i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (95.6 + 95.6i)T + 1.21e4iT^{2} \)
29 \( 1 + (-19.3 + 46.7i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 38.1T + 2.97e4T^{2} \)
37 \( 1 + (-227. + 94.2i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (279. + 279. i)T + 6.89e4iT^{2} \)
43 \( 1 + (112. + 270. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 321. iT - 1.03e5T^{2} \)
53 \( 1 + (52.5 + 126. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-332. + 137. i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-33.6 + 81.3i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (108. - 262. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (272. - 272. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-372. - 372. i)T + 3.89e5iT^{2} \)
79 \( 1 + 244. iT - 4.93e5T^{2} \)
83 \( 1 + (1.23e3 + 509. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-216. + 216. i)T - 7.04e5iT^{2} \)
97 \( 1 - 779.T + 9.12e5T^{2} \)
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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

−12.38969851654480563396813517099, −11.67015966963220243510186833524, −10.35864808115100862715180268149, −8.543011789417270528749345413033, −8.023871964943094707111404914278, −7.02357181896479198562938456627, −5.93257411378146003949990891621, −3.59840317213593662467685973785, −2.43895552529831119838205617271, −0.25860092197044917566514889711, 3.02972341445746951631121084117, 4.29215780127175837775594477426, 4.89041688155426450682271859371, 7.23812945286109845927929929278, 8.164913140679584993218751219746, 9.586868481660758193621137826069, 9.880484316835428759249170113752, 11.32234544367388131832291419892, 12.22724187619951841956552047633, 13.55138408874808440866810007621

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