Properties

Degree $2$
Conductor $128$
Sign $0.724 + 0.689i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.477 + 1.15i)3-s + (−16.3 − 6.77i)5-s + (18.0 + 18.0i)7-s + (17.9 − 17.9i)9-s + (20.1 − 48.5i)11-s + (37.8 − 15.6i)13-s − 22.0i·15-s − 53.0i·17-s + (32.4 − 13.4i)19-s + (−12.1 + 29.4i)21-s + (−32.1 + 32.1i)23-s + (132. + 132. i)25-s + (60.4 + 25.0i)27-s + (−52.0 − 125. i)29-s + 53.3·31-s + ⋯
L(s)  = 1  + (0.0919 + 0.221i)3-s + (−1.46 − 0.605i)5-s + (0.973 + 0.973i)7-s + (0.666 − 0.666i)9-s + (0.551 − 1.33i)11-s + (0.806 − 0.334i)13-s − 0.380i·15-s − 0.757i·17-s + (0.391 − 0.162i)19-s + (−0.126 + 0.305i)21-s + (−0.291 + 0.291i)23-s + (1.06 + 1.06i)25-s + (0.431 + 0.178i)27-s + (−0.333 − 0.804i)29-s + 0.309·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.724 + 0.689i$
Motivic weight: \(3\)
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.39144 - 0.556129i\)
\(L(\frac12)\) \(\approx\) \(1.39144 - 0.556129i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.477 - 1.15i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (16.3 + 6.77i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (-18.0 - 18.0i)T + 343iT^{2} \)
11 \( 1 + (-20.1 + 48.5i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-37.8 + 15.6i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 53.0iT - 4.91e3T^{2} \)
19 \( 1 + (-32.4 + 13.4i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (32.1 - 32.1i)T - 1.21e4iT^{2} \)
29 \( 1 + (52.0 + 125. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 53.3T + 2.97e4T^{2} \)
37 \( 1 + (57.3 + 23.7i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-240. + 240. i)T - 6.89e4iT^{2} \)
43 \( 1 + (56.3 - 135. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 314. iT - 1.03e5T^{2} \)
53 \( 1 + (177. - 428. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-133. - 55.4i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (191. + 462. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-55.4 - 133. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (191. + 191. i)T + 3.57e5iT^{2} \)
73 \( 1 + (175. - 175. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.22e3iT - 4.93e5T^{2} \)
83 \( 1 + (896. - 371. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-883. - 883. i)T + 7.04e5iT^{2} \)
97 \( 1 + 682.T + 9.12e5T^{2} \)
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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

−12.44417987623468795634425031705, −11.67403575362664197479235417166, −11.09917401789450833161291994787, −9.237171262458797408069763584345, −8.513976937661737988209599704431, −7.62334807777101286581989958385, −5.91138187762735490376268328289, −4.52887064716702348219985477034, −3.41918039628173905908900533059, −0.908750052461166322737982576038, 1.54973726867962116551284215516, 3.85726785194111673837458049178, 4.57189920109780568391977134292, 6.88126785762109837536974812482, 7.51888691036549507678670685477, 8.357430630040858868367184693193, 10.17013051984217131857129751951, 11.00428969464121679807024657638, 11.83077950347365791950200401429, 12.92503489909041265097553959156

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