Properties

Degree $2$
Conductor $128$
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

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.457 + 0.889i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.457 + 0.889i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.42042 - 0.866554i\)
\(L(\frac12)\) \(\approx\) \(1.42042 - 0.866554i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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