Properties

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

Origins

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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: $\chi_{128} (17, \cdot )$
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 \( 1 + 74.0T + 9.12e5T^{2} \)
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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

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

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