Properties

Degree $2$
Conductor $128$
Sign $-0.631 - 0.775i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.64 − 3.96i)3-s + (−11.8 − 4.89i)5-s + (5.11 + 5.11i)7-s + (6.09 − 6.09i)9-s + (−15.2 + 36.8i)11-s + (−73.4 + 30.4i)13-s + 54.7i·15-s + 66.8i·17-s + (37.0 − 15.3i)19-s + (11.8 − 28.6i)21-s + (−30.1 + 30.1i)23-s + (27.1 + 27.1i)25-s + (−141. − 58.4i)27-s + (64.4 + 155. i)29-s − 219.·31-s + ⋯
L(s)  = 1  + (−0.315 − 0.762i)3-s + (−1.05 − 0.437i)5-s + (0.276 + 0.276i)7-s + (0.225 − 0.225i)9-s + (−0.417 + 1.00i)11-s + (−1.56 + 0.649i)13-s + 0.943i·15-s + 0.954i·17-s + (0.447 − 0.185i)19-s + (0.123 − 0.297i)21-s + (−0.273 + 0.273i)23-s + (0.217 + 0.217i)25-s + (−1.00 − 0.416i)27-s + (0.412 + 0.996i)29-s − 1.26·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.631 - 0.775i$
Motivic weight: \(3\)
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0605407 + 0.127461i\)
\(L(\frac12)\) \(\approx\) \(0.0605407 + 0.127461i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.64 + 3.96i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (11.8 + 4.89i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (-5.11 - 5.11i)T + 343iT^{2} \)
11 \( 1 + (15.2 - 36.8i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (73.4 - 30.4i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 66.8iT - 4.91e3T^{2} \)
19 \( 1 + (-37.0 + 15.3i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (30.1 - 30.1i)T - 1.21e4iT^{2} \)
29 \( 1 + (-64.4 - 155. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 219.T + 2.97e4T^{2} \)
37 \( 1 + (286. + 118. i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-64.2 + 64.2i)T - 6.89e4iT^{2} \)
43 \( 1 + (-200. + 484. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 392. iT - 1.03e5T^{2} \)
53 \( 1 + (-107. + 258. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (237. + 98.4i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (43.9 + 106. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-333. - 804. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (387. + 387. i)T + 3.57e5iT^{2} \)
73 \( 1 + (518. - 518. i)T - 3.89e5iT^{2} \)
79 \( 1 + 214. iT - 4.93e5T^{2} \)
83 \( 1 + (436. - 180. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (877. + 877. i)T + 7.04e5iT^{2} \)
97 \( 1 - 43.7T + 9.12e5T^{2} \)
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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

−12.65458036990856923200142222859, −12.39991789284970927501560597990, −11.57760335923696891125400881085, −10.14417733493381074908690107995, −8.893226588052523692289301368651, −7.54308633704894610373723626223, −7.07021253518503445944163302761, −5.28741592460453145202817105999, −4.07603139360948242621022873607, −1.89517682334307004175253877322, 0.07297010082473211155855719257, 3.01882097334792524809252031941, 4.39120168745133616279494783693, 5.43582957627675415029356177423, 7.30960868375806876123959940842, 7.954146874708911226973586208373, 9.585179425992753781611789307872, 10.55832530113872449854539674336, 11.31787825335622440132416778334, 12.24542943602429506138405527322

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