Properties

Degree 2
Conductor $ 2^{7} $
Sign $0.457 - 0.889i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.91 − 3.91i)3-s + (−4.72 + 4.72i)5-s − 45.3·7-s + 50.3i·9-s + (110. + 110. i)11-s + (157. + 157. i)13-s + 36.9i·15-s − 378.·17-s + (203. − 203. i)19-s + (−177. + 177. i)21-s + 740.·23-s + 580. i·25-s + (514. + 514. i)27-s + (−82.6 − 82.6i)29-s + 286. i·31-s + ⋯
L(s)  = 1  + (0.434 − 0.434i)3-s + (−0.188 + 0.188i)5-s − 0.925·7-s + 0.621i·9-s + (0.910 + 0.910i)11-s + (0.929 + 0.929i)13-s + 0.164i·15-s − 1.31·17-s + (0.562 − 0.562i)19-s + (−0.402 + 0.402i)21-s + 1.39·23-s + 0.928i·25-s + (0.705 + 0.705i)27-s + (−0.0982 − 0.0982i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $0.457 - 0.889i$
motivic weight  =  \(4\)
character  :  $\chi_{128} (95, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :2),\ 0.457 - 0.889i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.42042 + 0.866554i\)
\(L(\frac12)\)  \(\approx\)  \(1.42042 + 0.866554i\)
\(L(3)\)   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 \( 1 + (-3.91 + 3.91i)T - 81iT^{2} \)
5 \( 1 + (4.72 - 4.72i)T - 625iT^{2} \)
7 \( 1 + 45.3T + 2.40e3T^{2} \)
11 \( 1 + (-110. - 110. i)T + 1.46e4iT^{2} \)
13 \( 1 + (-157. - 157. i)T + 2.85e4iT^{2} \)
17 \( 1 + 378.T + 8.35e4T^{2} \)
19 \( 1 + (-203. + 203. i)T - 1.30e5iT^{2} \)
23 \( 1 - 740.T + 2.79e5T^{2} \)
29 \( 1 + (82.6 + 82.6i)T + 7.07e5iT^{2} \)
31 \( 1 - 286. iT - 9.23e5T^{2} \)
37 \( 1 + (1.47e3 - 1.47e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + (366. + 366. i)T + 3.41e6iT^{2} \)
47 \( 1 - 751. iT - 4.87e6T^{2} \)
53 \( 1 + (-1.92e3 + 1.92e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (1.35e3 + 1.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-1.83e3 - 1.83e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-2.20e3 + 2.20e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 8.97e3T + 2.54e7T^{2} \)
73 \( 1 + 9.35e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.03e3 - 1.03e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 5.17e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.53e3T + 8.85e7T^{2} \)
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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

−13.14254694652712202741629313997, −11.78707458637571215598827857172, −10.86573675101625824664391575253, −9.432067355600792142744779327915, −8.728018262153114982260286100415, −7.12523026654152803879928616885, −6.62238091740757854082467432146, −4.69836546758112868691661178699, −3.23216144555519374258734826568, −1.65936930420181572104160360815, 0.69591804884145730657163591976, 3.13612307607975077017458026032, 3.97656530663399418593347847996, 5.84377862621651189388376926560, 6.84498951677803116611357708637, 8.598912369640877782732140624776, 9.077075110646706086501720892937, 10.30445247826839011223181517322, 11.38864488515808634496167020501, 12.56034041949247003301341775031

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