Properties

Degree 2
Conductor $ 2^{7} $
Sign $-0.851 - 0.524i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.63 − 4.63i)3-s + (29.2 + 29.2i)5-s − 59.6·7-s − 38.0i·9-s + (−18.0 + 18.0i)11-s + (−50.7 + 50.7i)13-s − 270. i·15-s − 223.·17-s + (14.7 + 14.7i)19-s + (276. + 276. i)21-s − 739.·23-s + 1.08e3i·25-s + (−551. + 551. i)27-s + (−938. + 938. i)29-s + 938. i·31-s + ⋯
L(s)  = 1  + (−0.515 − 0.515i)3-s + (1.16 + 1.16i)5-s − 1.21·7-s − 0.469i·9-s + (−0.149 + 0.149i)11-s + (−0.300 + 0.300i)13-s − 1.20i·15-s − 0.774·17-s + (0.0408 + 0.0408i)19-s + (0.626 + 0.626i)21-s − 1.39·23-s + 1.72i·25-s + (−0.756 + 0.756i)27-s + (−1.11 + 1.11i)29-s + 0.976i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $-0.851 - 0.524i$
motivic weight  =  \(4\)
character  :  $\chi_{128} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :2),\ -0.851 - 0.524i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.117794 + 0.415962i\)
\(L(\frac12)\)  \(\approx\)  \(0.117794 + 0.415962i\)
\(L(3)\)   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 \( 1 + (4.63 + 4.63i)T + 81iT^{2} \)
5 \( 1 + (-29.2 - 29.2i)T + 625iT^{2} \)
7 \( 1 + 59.6T + 2.40e3T^{2} \)
11 \( 1 + (18.0 - 18.0i)T - 1.46e4iT^{2} \)
13 \( 1 + (50.7 - 50.7i)T - 2.85e4iT^{2} \)
17 \( 1 + 223.T + 8.35e4T^{2} \)
19 \( 1 + (-14.7 - 14.7i)T + 1.30e5iT^{2} \)
23 \( 1 + 739.T + 2.79e5T^{2} \)
29 \( 1 + (938. - 938. i)T - 7.07e5iT^{2} \)
31 \( 1 - 938. iT - 9.23e5T^{2} \)
37 \( 1 + (263. + 263. i)T + 1.87e6iT^{2} \)
41 \( 1 - 248. iT - 2.82e6T^{2} \)
43 \( 1 + (-1.03e3 + 1.03e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 2.01e3iT - 4.87e6T^{2} \)
53 \( 1 + (833. + 833. i)T + 7.89e6iT^{2} \)
59 \( 1 + (2.22e3 - 2.22e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-341. + 341. i)T - 1.38e7iT^{2} \)
67 \( 1 + (-4.84e3 - 4.84e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 4.18e3T + 2.54e7T^{2} \)
73 \( 1 + 9.07e3iT - 2.83e7T^{2} \)
79 \( 1 + 735. iT - 3.89e7T^{2} \)
83 \( 1 + (-1.44e3 - 1.44e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.07e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.52e3T + 8.85e7T^{2} \)
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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

−13.04337276552230574813275571752, −12.17095821325186965330150302098, −10.89366032364436719629283882571, −9.977982073491203186661309803249, −9.175271463370932557492348637940, −7.09943836234043043507240858683, −6.53429024408288252148542384815, −5.67748178742977164595775841292, −3.44164940185344518673698166186, −2.04063070418753170461591560135, 0.17491578382766491059102145480, 2.21701928078806766835142824230, 4.27629832461966334730737585261, 5.50894796676766502277863581628, 6.22050893374087887026501686549, 8.050152981934790991719778205600, 9.528669764683825544916629033439, 9.783078150896455217902540418910, 11.05274116441837768228067013393, 12.44499242619513407818966742899

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