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: $\chi_{128} (95, \cdot )$
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

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

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