Properties

Label 2-2e7-4.3-c4-0-9
Degree $2$
Conductor $128$
Sign $i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7.95i·3-s − 46.0·5-s + 57.5i·7-s + 17.7·9-s − 181. i·11-s − 1.33·13-s − 366. i·15-s − 62.6·17-s − 340. i·19-s − 457.·21-s − 447. i·23-s + 1.49e3·25-s + 785. i·27-s − 497.·29-s − 444. i·31-s + ⋯
L(s)  = 1  + 0.883i·3-s − 1.84·5-s + 1.17i·7-s + 0.219·9-s − 1.50i·11-s − 0.00787·13-s − 1.62i·15-s − 0.216·17-s − 0.942i·19-s − 1.03·21-s − 0.845i·23-s + 2.39·25-s + 1.07i·27-s − 0.591·29-s − 0.462i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.279104 - 0.279104i\)
\(L(\frac12)\) \(\approx\) \(0.279104 - 0.279104i\)
\(L(3)\) 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 \( 1 - 7.95iT - 81T^{2} \)
5 \( 1 + 46.0T + 625T^{2} \)
7 \( 1 - 57.5iT - 2.40e3T^{2} \)
11 \( 1 + 181. iT - 1.46e4T^{2} \)
13 \( 1 + 1.33T + 2.85e4T^{2} \)
17 \( 1 + 62.6T + 8.35e4T^{2} \)
19 \( 1 + 340. iT - 1.30e5T^{2} \)
23 \( 1 + 447. iT - 2.79e5T^{2} \)
29 \( 1 + 497.T + 7.07e5T^{2} \)
31 \( 1 + 444. iT - 9.23e5T^{2} \)
37 \( 1 + 687.T + 1.87e6T^{2} \)
41 \( 1 + 3.07e3T + 2.82e6T^{2} \)
43 \( 1 + 307. iT - 3.41e6T^{2} \)
47 \( 1 + 4.04e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.97e3T + 7.89e6T^{2} \)
59 \( 1 - 2.87e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.53e3T + 1.38e7T^{2} \)
67 \( 1 - 3.96e3iT - 2.01e7T^{2} \)
71 \( 1 - 910. iT - 2.54e7T^{2} \)
73 \( 1 + 4.40e3T + 2.83e7T^{2} \)
79 \( 1 + 3.39e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.38e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.52e3T + 6.27e7T^{2} \)
97 \( 1 + 9.68e3T + 8.85e7T^{2} \)
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

−12.03789528387323464344220348195, −11.42283952941994371142041671565, −10.51808375644252358076058568797, −8.885832927718517112427222151635, −8.439044410176831508738591985833, −6.98859463466360932806466084785, −5.33135325551397390549649205437, −4.12668559076560754871851564261, −3.09235504121407914201739380025, −0.17803050075216807362781695384, 1.40816662538624578850880375569, 3.68823581047874846273907781867, 4.58576668412179589228782500321, 6.92334815481932108174987205538, 7.38957944258554585555548330264, 8.132525247855342748138763867003, 9.894576490420777640112381019454, 11.01856183372077025488574068452, 12.12391406794645191891650151817, 12.61231694005436845457669512069

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