Properties

Degree $2$
Conductor $128$
Sign $-0.0331 - 0.999i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 + 4.59i)3-s + (0.188 + 0.0782i)5-s + (11.4 + 11.4i)7-s + (1.63 − 1.63i)9-s + (−18.6 + 45.0i)11-s + (18.9 − 7.84i)13-s + 1.01i·15-s + 85.7i·17-s + (−110. + 45.8i)19-s + (−30.6 + 74.0i)21-s + (74.2 − 74.2i)23-s + (−88.3 − 88.3i)25-s + (134. + 55.7i)27-s + (64.4 + 155. i)29-s − 36.6·31-s + ⋯
L(s)  = 1  + (0.365 + 0.883i)3-s + (0.0168 + 0.00699i)5-s + (0.616 + 0.616i)7-s + (0.0603 − 0.0603i)9-s + (−0.511 + 1.23i)11-s + (0.404 − 0.167i)13-s + 0.0174i·15-s + 1.22i·17-s + (−1.33 + 0.553i)19-s + (−0.318 + 0.769i)21-s + (0.672 − 0.672i)23-s + (−0.706 − 0.706i)25-s + (0.958 + 0.397i)27-s + (0.412 + 0.996i)29-s − 0.212·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0331 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.0331 - 0.999i$
Motivic weight: \(3\)
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.0331 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.27516 + 1.31820i\)
\(L(\frac12)\) \(\approx\) \(1.27516 + 1.31820i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.90 - 4.59i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-0.188 - 0.0782i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (-11.4 - 11.4i)T + 343iT^{2} \)
11 \( 1 + (18.6 - 45.0i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-18.9 + 7.84i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 85.7iT - 4.91e3T^{2} \)
19 \( 1 + (110. - 45.8i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-74.2 + 74.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (-64.4 - 155. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 36.6T + 2.97e4T^{2} \)
37 \( 1 + (-313. - 129. i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-196. + 196. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-20.8 + 50.4i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 508. iT - 1.03e5T^{2} \)
53 \( 1 + (-73.8 + 178. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (40.9 + 16.9i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (324. + 784. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-49.4 - 119. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-362. - 362. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-239. + 239. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.01e3iT - 4.93e5T^{2} \)
83 \( 1 + (231. - 95.7i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-1.10e3 - 1.10e3i)T + 7.04e5iT^{2} \)
97 \( 1 + 74.0T + 9.12e5T^{2} \)
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

−12.94366631001194732354272994295, −12.25395673756513793609300008288, −10.74566405851278800713494910635, −10.12123299024643732960135869726, −8.895323430162329677810991516612, −8.066617687620739486167449188354, −6.45035973257422418289186298816, −4.95029173298017951998822856723, −3.92653234028722414243213480280, −2.11389588502017984274730546734, 0.977646890635642344000520836352, 2.66221738781942850247375643248, 4.47254797214593633408702471004, 6.05055176626518196854562109314, 7.38623098585951030429036938292, 8.064221669884563410515948554349, 9.259820843201130213109255458683, 10.83176165523244886067602871269, 11.43269149071700205815347488796, 12.97129969434661493559894083885

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