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Properties

Label 2-2e8-32.3-c2-0-13
Degree $2$
Conductor $256$
Sign $-0.979 + 0.201i$
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  + (1.10 − 2.67i)3-s + (−2.95 − 7.13i)5-s + (−4.18 − 4.18i)7-s + (0.437 + 0.437i)9-s + (−1.42 − 3.44i)11-s + (−8.39 + 20.2i)13-s − 22.3·15-s + 1.73i·17-s + (−14.2 − 5.90i)19-s + (−15.8 + 6.55i)21-s + (15.1 − 15.1i)23-s + (−24.5 + 24.5i)25-s + (25.7 − 10.6i)27-s + (6.74 + 2.79i)29-s − 31.1i·31-s + ⋯
L(s)  = 1  + (0.369 − 0.891i)3-s + (−0.591 − 1.42i)5-s + (−0.597 − 0.597i)7-s + (0.0486 + 0.0486i)9-s + (−0.129 − 0.313i)11-s + (−0.646 + 1.55i)13-s − 1.49·15-s + 0.101i·17-s + (−0.749 − 0.310i)19-s + (−0.753 + 0.312i)21-s + (0.658 − 0.658i)23-s + (−0.980 + 0.980i)25-s + (0.952 − 0.394i)27-s + (0.232 + 0.0962i)29-s − 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.979 + 0.201i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ -0.979 + 0.201i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.109139 - 1.07462i\)
\(L(\frac12)\) \(\approx\) \(0.109139 - 1.07462i\)
\(L(2)\) 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 \( 1 + (-1.10 + 2.67i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (2.95 + 7.13i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (4.18 + 4.18i)T + 49iT^{2} \)
11 \( 1 + (1.42 + 3.44i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (8.39 - 20.2i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 1.73iT - 289T^{2} \)
19 \( 1 + (14.2 + 5.90i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-15.1 + 15.1i)T - 529iT^{2} \)
29 \( 1 + (-6.74 - 2.79i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + (5.30 + 12.7i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (18.5 + 18.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (31.0 + 75.0i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 16.2T + 2.20e3T^{2} \)
53 \( 1 + (-29.0 + 12.0i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (34.1 - 14.1i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-68.7 - 28.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (10.5 - 25.3i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (32.2 + 32.2i)T + 5.04e3iT^{2} \)
73 \( 1 + (28.5 + 28.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 22.4T + 6.24e3T^{2} \)
83 \( 1 + (-123. - 51.0i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-61.0 + 61.0i)T - 7.92e3iT^{2} \)
97 \( 1 + 69.9T + 9.40e3T^{2} \)
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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

−11.70718112003428842969974521001, −10.34750515518507377198003460442, −9.088888994629753835328830300899, −8.469646548467853258579438656807, −7.36916217234452000674171796038, −6.61975575749654049310952012637, −4.91132073282577216314141247927, −3.98910029595393127160440024207, −2.05021516658095293586463334182, −0.52407100432778070676759316740, 2.87609234852196291079467547518, 3.39984628666053482672575355241, 4.88435729678180026983976564231, 6.31046262985501886204873402997, 7.30726273039851998670960024048, 8.371452814652448720772422401121, 9.670128717586756978331482227353, 10.24131107249673844888243377571, 11.02650048526582054587354709902, 12.20699144283304912124330285794

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