Properties

Degree 2
Conductor $ 2^{7} $
Sign $0.0172 + 0.999i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.5 − 11.5i)3-s + (14.6 − 14.6i)5-s + 24.0·7-s − 184. i·9-s + (61.7 + 61.7i)11-s + (37.5 + 37.5i)13-s − 336. i·15-s + 96.8·17-s + (−156. + 156. i)19-s + (276. − 276. i)21-s − 959.·23-s + 198. i·25-s + (−1.19e3 − 1.19e3i)27-s + (350. + 350. i)29-s − 237. i·31-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)3-s + (0.584 − 0.584i)5-s + 0.490·7-s − 2.27i·9-s + (0.510 + 0.510i)11-s + (0.222 + 0.222i)13-s − 1.49i·15-s + 0.335·17-s + (−0.434 + 0.434i)19-s + (0.627 − 0.627i)21-s − 1.81·23-s + 0.317i·25-s + (−1.63 − 1.63i)27-s + (0.416 + 0.416i)29-s − 0.247i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0172 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0172 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $0.0172 + 0.999i$
motivic weight  =  \(4\)
character  :  $\chi_{128} (95, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :2),\ 0.0172 + 0.999i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.21371 - 2.17582i\)
\(L(\frac12)\)  \(\approx\)  \(2.21371 - 2.17582i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-11.5 + 11.5i)T - 81iT^{2} \)
5 \( 1 + (-14.6 + 14.6i)T - 625iT^{2} \)
7 \( 1 - 24.0T + 2.40e3T^{2} \)
11 \( 1 + (-61.7 - 61.7i)T + 1.46e4iT^{2} \)
13 \( 1 + (-37.5 - 37.5i)T + 2.85e4iT^{2} \)
17 \( 1 - 96.8T + 8.35e4T^{2} \)
19 \( 1 + (156. - 156. i)T - 1.30e5iT^{2} \)
23 \( 1 + 959.T + 2.79e5T^{2} \)
29 \( 1 + (-350. - 350. i)T + 7.07e5iT^{2} \)
31 \( 1 + 237. iT - 9.23e5T^{2} \)
37 \( 1 + (-560. + 560. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (-206. - 206. i)T + 3.41e6iT^{2} \)
47 \( 1 - 1.59e3iT - 4.87e6T^{2} \)
53 \( 1 + (-2.23e3 + 2.23e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (-2.35e3 - 2.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-4.44e3 - 4.44e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (3.99e3 - 3.99e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 4.92e3T + 2.54e7T^{2} \)
73 \( 1 + 2.65e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.79e3iT - 3.89e7T^{2} \)
83 \( 1 + (228. - 228. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.10e4T + 8.85e7T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.58925558982638737977843323544, −11.80538734026734819496650009734, −9.951161698333174907166188838777, −8.932263534056536750438736173912, −8.150593476440699335408676188333, −7.12976103136789684375350530217, −5.89185299690616742342414840192, −3.97408837517738363830008522475, −2.20873140985842180013465386818, −1.31475066414417438437216437719, 2.20385623016150239100261330432, 3.44136663753688172213832381879, 4.59990317503054867646383229008, 6.14511011209659226627670899086, 7.929031009130063665809162964930, 8.719580292578451076237577872347, 9.854060817289011471244123166168, 10.42987487672394987129211687606, 11.59164650233085484895202005175, 13.37148700784001980653462510814

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