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Properties

Label 2-2e7-16.3-c4-0-11
Degree $2$
Conductor $128$
Sign $-0.851 + 0.524i$
Analytic cond. $13.2313$
Root an. cond. $3.63749$
Motivic weight $4$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -0.851 + 0.524i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.117794 - 0.415962i\)
\(L(\frac12)\) \(\approx\) \(0.117794 - 0.415962i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

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

Imaginary part of the first few zeros on the critical line

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

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