Properties

Degree $2$
Conductor $128$
Sign $-0.258 - 0.966i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.54 − 5.54i)3-s + (−21.7 − 21.7i)5-s − 6.62·7-s − 19.6i·9-s + (90.9 − 90.9i)11-s + (−221. + 221. i)13-s + 240. i·15-s − 132.·17-s + (402. + 402. i)19-s + (36.6 + 36.6i)21-s + 27.5·23-s + 320. i·25-s + (−557. + 557. i)27-s + (−174. + 174. i)29-s + 1.08e3i·31-s + ⋯
L(s)  = 1  + (−0.615 − 0.615i)3-s + (−0.869 − 0.869i)5-s − 0.135·7-s − 0.242i·9-s + (0.752 − 0.752i)11-s + (−1.31 + 1.31i)13-s + 1.07i·15-s − 0.458·17-s + (1.11 + 1.11i)19-s + (0.0831 + 0.0831i)21-s + 0.0519·23-s + 0.512i·25-s + (−0.764 + 0.764i)27-s + (−0.207 + 0.207i)29-s + 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.258 - 0.966i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -0.258 - 0.966i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0679138 + 0.0884764i\)
\(L(\frac12)\) \(\approx\) \(0.0679138 + 0.0884764i\)
\(L(3)\) 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 \( 1 + (5.54 + 5.54i)T + 81iT^{2} \)
5 \( 1 + (21.7 + 21.7i)T + 625iT^{2} \)
7 \( 1 + 6.62T + 2.40e3T^{2} \)
11 \( 1 + (-90.9 + 90.9i)T - 1.46e4iT^{2} \)
13 \( 1 + (221. - 221. i)T - 2.85e4iT^{2} \)
17 \( 1 + 132.T + 8.35e4T^{2} \)
19 \( 1 + (-402. - 402. i)T + 1.30e5iT^{2} \)
23 \( 1 - 27.5T + 2.79e5T^{2} \)
29 \( 1 + (174. - 174. i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 + (553. + 553. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (17.8 - 17.8i)T - 3.41e6iT^{2} \)
47 \( 1 + 2.26e3iT - 4.87e6T^{2} \)
53 \( 1 + (-822. - 822. i)T + 7.89e6iT^{2} \)
59 \( 1 + (-972. + 972. i)T - 1.21e7iT^{2} \)
61 \( 1 + (-2.05e3 + 2.05e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (4.61e3 + 4.61e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 3.10e3T + 2.54e7T^{2} \)
73 \( 1 + 723. iT - 2.83e7T^{2} \)
79 \( 1 - 3.41e3iT - 3.89e7T^{2} \)
83 \( 1 + (-161. - 161. i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 8.26e3T + 8.85e7T^{2} \)
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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

−12.50796311137591251369280296135, −12.00031448857989498240111486328, −11.38299415777482512385902019723, −9.638066046967371948692635970059, −8.691656338329548305127196243927, −7.40617274414412821509835547581, −6.43630804248601222233711421257, −5.02426396578805259490191492777, −3.71243605080742447824123777291, −1.34469767202634270581938669457, 0.05395311701507841518837322622, 2.78274516930953647343859368252, 4.26690853838354810058213686692, 5.38240600311613595704135025001, 7.00258007191260442316354479038, 7.73115604712568516298142759366, 9.492762317498940406094133116151, 10.35965120533682101056278081851, 11.31437657706937049536312153643, 12.00819432058152465268662513442

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