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Properties

Label 2-2e8-32.11-c2-0-9
Degree $2$
Conductor $256$
Sign $0.996 - 0.0818i$
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  + (1.31 + 3.18i)3-s + (0.659 − 1.59i)5-s + (9.54 − 9.54i)7-s + (−2.03 + 2.03i)9-s + (3.96 − 9.57i)11-s + (−1.91 − 4.63i)13-s + 5.93·15-s + 15.3i·17-s + (−0.827 + 0.342i)19-s + (42.9 + 17.8i)21-s + (−12.9 − 12.9i)23-s + (15.5 + 15.5i)25-s + (19.5 + 8.07i)27-s + (−23.7 + 9.85i)29-s + 25.1i·31-s + ⋯
L(s)  = 1  + (0.439 + 1.06i)3-s + (0.131 − 0.318i)5-s + (1.36 − 1.36i)7-s + (−0.225 + 0.225i)9-s + (0.360 − 0.870i)11-s + (−0.147 − 0.356i)13-s + 0.395·15-s + 0.900i·17-s + (−0.0435 + 0.0180i)19-s + (2.04 + 0.847i)21-s + (−0.561 − 0.561i)23-s + (0.623 + 0.623i)25-s + (0.722 + 0.299i)27-s + (−0.820 + 0.339i)29-s + 0.811i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.996 - 0.0818i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ 0.996 - 0.0818i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.14573 + 0.0879350i\)
\(L(\frac12)\) \(\approx\) \(2.14573 + 0.0879350i\)
\(L(2)\) 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 \( 1 + (-1.31 - 3.18i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.659 + 1.59i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-9.54 + 9.54i)T - 49iT^{2} \)
11 \( 1 + (-3.96 + 9.57i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (1.91 + 4.63i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 15.3iT - 289T^{2} \)
19 \( 1 + (0.827 - 0.342i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (12.9 + 12.9i)T + 529iT^{2} \)
29 \( 1 + (23.7 - 9.85i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 25.1iT - 961T^{2} \)
37 \( 1 + (13.6 - 32.8i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (32.9 - 32.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (-17.9 + 43.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 20.1T + 2.20e3T^{2} \)
53 \( 1 + (35.0 + 14.5i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-60.6 - 25.1i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-27.9 + 11.5i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (1.13 + 2.73i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (45.6 - 45.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (29.1 - 29.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.27T + 6.24e3T^{2} \)
83 \( 1 + (56.7 - 23.5i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (44.5 + 44.5i)T + 7.92e3iT^{2} \)
97 \( 1 + 106.T + 9.40e3T^{2} \)
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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

−11.49444797474442959827960473651, −10.64619498439276320737316103157, −10.12026530317079805985728290102, −8.785301270621915349766764567924, −8.202560014235174994234786661003, −6.94100531608035433105993637453, −5.31476771808270023631755186539, −4.34228862670419362314182027859, −3.50913208968424842571567519519, −1.30791061466395159305546336454, 1.76298155179928956060897558790, 2.46561553595817957157065252579, 4.56888473730661399382418870219, 5.73981158134744620618611407564, 7.03112878912620129582044437085, 7.78594491931550640617913996098, 8.715940972987346834950371240676, 9.650750348308037482604571416740, 11.13608248735557313051893867302, 11.95145636044348596144677112701

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