Properties

Degree $2$
Conductor $128$
Sign $0.410 - 0.911i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.18 + 3.80i)3-s + (1.04 + 2.51i)5-s + (−16.1 + 16.1i)7-s + (50.8 + 50.8i)9-s + (−3.72 + 1.54i)11-s + (9.23 − 22.3i)13-s + 27.0i·15-s − 4.95i·17-s + (26.0 − 62.8i)19-s + (−209. + 86.6i)21-s + (−82.8 − 82.8i)23-s + (83.1 − 83.1i)25-s + (170. + 412. i)27-s + (150. + 62.3i)29-s + 141.·31-s + ⋯
L(s)  = 1  + (1.76 + 0.732i)3-s + (0.0932 + 0.225i)5-s + (−0.869 + 0.869i)7-s + (1.88 + 1.88i)9-s + (−0.102 + 0.0422i)11-s + (0.197 − 0.475i)13-s + 0.466i·15-s − 0.0706i·17-s + (0.314 − 0.758i)19-s + (−2.17 + 0.900i)21-s + (−0.750 − 0.750i)23-s + (0.665 − 0.665i)25-s + (1.21 + 2.93i)27-s + (0.963 + 0.399i)29-s + 0.820·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.410 - 0.911i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.410 - 0.911i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.25655 + 1.45805i\)
\(L(\frac12)\) \(\approx\) \(2.25655 + 1.45805i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.18 - 3.80i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.04 - 2.51i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (16.1 - 16.1i)T - 343iT^{2} \)
11 \( 1 + (3.72 - 1.54i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-9.23 + 22.3i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 4.95iT - 4.91e3T^{2} \)
19 \( 1 + (-26.0 + 62.8i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (82.8 + 82.8i)T + 1.21e4iT^{2} \)
29 \( 1 + (-150. - 62.3i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 141.T + 2.97e4T^{2} \)
37 \( 1 + (1.05 + 2.55i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-8.70 - 8.70i)T + 6.89e4iT^{2} \)
43 \( 1 + (290. - 120. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 450. iT - 1.03e5T^{2} \)
53 \( 1 + (114. - 47.4i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (124. + 300. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (223. + 92.7i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (204. + 84.5i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-606. + 606. i)T - 3.57e5iT^{2} \)
73 \( 1 + (531. + 531. i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.12e3iT - 4.93e5T^{2} \)
83 \( 1 + (-118. + 286. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-191. + 191. i)T - 7.04e5iT^{2} \)
97 \( 1 + 38.4T + 9.12e5T^{2} \)
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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

−13.31238261675017160435008259627, −12.30413820106701169690122125212, −10.52364810805054695998503187390, −9.766106068105822930220801094339, −8.846056896033949847746456282586, −8.091182339336842198577124990198, −6.59941766521088846648749256278, −4.78551437736391544201572480698, −3.27347803586278890304994797093, −2.48725503346942006196700459560, 1.35859206581633489657844433672, 3.00981516615191235479891431844, 4.04935213016304872334118837838, 6.44119947862759220575032294340, 7.41459816554313117358925659209, 8.343236336829672940206652449918, 9.415158587946939305768994274604, 10.16657305556175211976143230284, 12.04877492921935991408942391818, 13.05402713839769880295765513767

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