Properties

Degree $2$
Conductor $128$
Sign $0.574 + 0.818i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.527 − 1.27i)3-s + (−0.642 + 1.55i)5-s + (4.95 − 4.95i)7-s + (5.01 − 5.01i)9-s + (4.27 − 10.3i)11-s + (1.68 + 4.06i)13-s + 2.31·15-s − 28.6i·17-s + (−17.5 + 7.26i)19-s + (−8.91 − 3.69i)21-s + (24.3 + 24.3i)23-s + (15.6 + 15.6i)25-s + (−20.5 − 8.49i)27-s + (8.57 − 3.55i)29-s + 5.73i·31-s + ⋯
L(s)  = 1  + (−0.175 − 0.424i)3-s + (−0.128 + 0.310i)5-s + (0.707 − 0.707i)7-s + (0.557 − 0.557i)9-s + (0.388 − 0.937i)11-s + (0.129 + 0.312i)13-s + 0.154·15-s − 1.68i·17-s + (−0.923 + 0.382i)19-s + (−0.424 − 0.175i)21-s + (1.05 + 1.05i)23-s + (0.627 + 0.627i)25-s + (−0.759 − 0.314i)27-s + (0.295 − 0.122i)29-s + 0.184i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.574 + 0.818i$
Motivic weight: \(2\)
Character: $\chi_{128} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.574 + 0.818i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24967 - 0.649319i\)
\(L(\frac12)\) \(\approx\) \(1.24967 - 0.649319i\)
\(L(2)\) 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 \( 1 + (0.527 + 1.27i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.642 - 1.55i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-4.95 + 4.95i)T - 49iT^{2} \)
11 \( 1 + (-4.27 + 10.3i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.68 - 4.06i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 28.6iT - 289T^{2} \)
19 \( 1 + (17.5 - 7.26i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-24.3 - 24.3i)T + 529iT^{2} \)
29 \( 1 + (-8.57 + 3.55i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 5.73iT - 961T^{2} \)
37 \( 1 + (26.1 - 63.0i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (14.2 - 14.2i)T - 1.68e3iT^{2} \)
43 \( 1 + (-10.1 + 24.4i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 57.9T + 2.20e3T^{2} \)
53 \( 1 + (46.3 + 19.2i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-27.6 - 11.4i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-76.3 + 31.6i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-36.1 - 87.3i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-5.39 + 5.39i)T - 5.04e3iT^{2} \)
73 \( 1 + (25.4 - 25.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 50.1T + 6.24e3T^{2} \)
83 \( 1 + (-100. + 41.7i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-10.6 - 10.6i)T + 7.92e3iT^{2} \)
97 \( 1 + 14.3T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.06884583412243150146696486512, −11.68185079542069315123208591959, −11.18308563409602577821904922460, −9.858574833563558901653082800944, −8.632177019954520776652633424834, −7.32202758023659292025956003955, −6.57748501152952072945976424800, −4.92112216399616530112875882028, −3.43137841843662130645224892381, −1.16537450589595833840438153490, 1.97258790392094639975171797338, 4.22756411775377226405040044133, 5.11975652291653817141832164768, 6.63025892600019314681737897464, 8.099044422412105057866094050728, 8.950632272318277355068358153880, 10.33500146908359940451065177545, 11.02502556496421657688894214984, 12.44925308224432261995486561889, 12.89942055382767495821729201146

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