Properties

Degree 2
Conductor $ 2^{7} $
Sign $-0.432 + 0.901i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.42 − 9.42i)3-s + (2.84 + 2.84i)5-s + 76.7·7-s + 96.6i·9-s + (121. − 121. i)11-s + (−27.1 + 27.1i)13-s − 53.6i·15-s − 88.0·17-s + (−261. − 261. i)19-s + (−723. − 723. i)21-s − 93.4·23-s − 608. i·25-s + (147. − 147. i)27-s + (−272. + 272. i)29-s − 1.23e3i·31-s + ⋯
L(s)  = 1  + (−1.04 − 1.04i)3-s + (0.113 + 0.113i)5-s + 1.56·7-s + 1.19i·9-s + (1.00 − 1.00i)11-s + (−0.160 + 0.160i)13-s − 0.238i·15-s − 0.304·17-s + (−0.723 − 0.723i)19-s + (−1.64 − 1.64i)21-s − 0.176·23-s − 0.974i·25-s + (0.202 − 0.202i)27-s + (−0.324 + 0.324i)29-s − 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $-0.432 + 0.901i$
motivic weight  =  \(4\)
character  :  $\chi_{128} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :2),\ -0.432 + 0.901i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.709564 - 1.12716i\)
\(L(\frac12)\)  \(\approx\)  \(0.709564 - 1.12716i\)
\(L(3)\)   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 \( 1 + (9.42 + 9.42i)T + 81iT^{2} \)
5 \( 1 + (-2.84 - 2.84i)T + 625iT^{2} \)
7 \( 1 - 76.7T + 2.40e3T^{2} \)
11 \( 1 + (-121. + 121. i)T - 1.46e4iT^{2} \)
13 \( 1 + (27.1 - 27.1i)T - 2.85e4iT^{2} \)
17 \( 1 + 88.0T + 8.35e4T^{2} \)
19 \( 1 + (261. + 261. i)T + 1.30e5iT^{2} \)
23 \( 1 + 93.4T + 2.79e5T^{2} \)
29 \( 1 + (272. - 272. i)T - 7.07e5iT^{2} \)
31 \( 1 + 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + (-1.04e3 - 1.04e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 915. iT - 2.82e6T^{2} \)
43 \( 1 + (-1.11e3 + 1.11e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.72e3iT - 4.87e6T^{2} \)
53 \( 1 + (734. + 734. i)T + 7.89e6iT^{2} \)
59 \( 1 + (1.20e3 - 1.20e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (580. - 580. i)T - 1.38e7iT^{2} \)
67 \( 1 + (1.48e3 + 1.48e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 5.57e3T + 2.54e7T^{2} \)
73 \( 1 + 6.61e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.55e3 + 2.55e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.71e3T + 8.85e7T^{2} \)
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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

−12.00866140649145787870082324844, −11.45558592283156759362315090631, −10.74748682522092867009367852933, −8.897062746475311864435460326129, −7.83821040539931494895413579649, −6.66850443047989650100994468659, −5.75820278745101047311481625152, −4.43142264842906032293782355629, −1.96843901793091553169532659105, −0.68472825190667696835378444408, 1.58898800433732612502773552420, 4.19570725022014929573627267819, 4.86047762477464586699907536942, 6.02554472603940684030886366223, 7.55406738021391359742102578925, 8.959462848016159626313386039119, 10.03103642509175481121637925260, 11.01399446080414315826690151568, 11.64029160835191246524945840093, 12.64415378039170035825904993422

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