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Properties

Label 2-2e8-32.19-c2-0-7
Degree $2$
Conductor $256$
Sign $0.766 - 0.642i$
Analytic cond. $6.97549$
Root an. cond. $2.64111$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.35 + 1.80i)3-s + (2.81 − 1.16i)5-s + (6.23 + 6.23i)7-s + (9.32 + 9.32i)9-s + (−8.06 + 3.33i)11-s + (−13.3 − 5.51i)13-s + 14.3·15-s + 4.56i·17-s + (13.4 − 32.4i)19-s + (15.8 + 38.3i)21-s + (6.75 − 6.75i)23-s + (−11.1 + 11.1i)25-s + (7.53 + 18.1i)27-s + (0.266 − 0.643i)29-s − 0.326i·31-s + ⋯
L(s)  = 1  + (1.45 + 0.600i)3-s + (0.563 − 0.233i)5-s + (0.890 + 0.890i)7-s + (1.03 + 1.03i)9-s + (−0.732 + 0.303i)11-s + (−1.02 − 0.424i)13-s + 0.957·15-s + 0.268i·17-s + (0.707 − 1.70i)19-s + (0.756 + 1.82i)21-s + (0.293 − 0.293i)23-s + (−0.444 + 0.444i)25-s + (0.279 + 0.674i)27-s + (0.00918 − 0.0221i)29-s − 0.0105i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.61651 + 0.951089i\)
\(L(\frac12)\) \(\approx\) \(2.61651 + 0.951089i\)
\(L(2)\) 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 \( 1 + (-4.35 - 1.80i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-2.81 + 1.16i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-6.23 - 6.23i)T + 49iT^{2} \)
11 \( 1 + (8.06 - 3.33i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (13.3 + 5.51i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 4.56iT - 289T^{2} \)
19 \( 1 + (-13.4 + 32.4i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-6.75 + 6.75i)T - 529iT^{2} \)
29 \( 1 + (-0.266 + 0.643i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 0.326iT - 961T^{2} \)
37 \( 1 + (31.5 - 13.0i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-15.7 - 15.7i)T + 1.68e3iT^{2} \)
43 \( 1 + (-4.83 + 2.00i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 49.7T + 2.20e3T^{2} \)
53 \( 1 + (4.45 + 10.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-13.1 - 31.6i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-35.4 + 85.4i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (41.3 + 17.1i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (37.6 + 37.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (52.2 + 52.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 26.9T + 6.24e3T^{2} \)
83 \( 1 + (-10.6 + 25.6i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (103. - 103. i)T - 7.92e3iT^{2} \)
97 \( 1 - 77.9T + 9.40e3T^{2} \)
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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

−11.96130659455462487771445349509, −10.71768149172646831558611259450, −9.673692328116229457888468239650, −9.067881316419268321480229869115, −8.207620041363355007034752129710, −7.31396290550847026485946836086, −5.39046883313528499826719448872, −4.67562310568909657973568922229, −2.93127540889456152899364363610, −2.12971520712858826062356228717, 1.57457879015852406943255860007, 2.68939228188192150214757984624, 4.04587020722894466534788493849, 5.55026000971514498698623904728, 7.22688600798160085351314406590, 7.66404722517679842610223169301, 8.603615699875624234182779937632, 9.751495807123239517695061779088, 10.47434368556515102442300854214, 11.82509797630460906975132964901

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