↑

Properties

Label 2-2e8-32.3-c2-0-3
Degree $2$
Conductor $256$
Sign $-0.591 - 0.806i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.936 + 2.26i)3-s + (3.18 + 7.68i)5-s + (3.67 + 3.67i)7-s + (2.12 + 2.12i)9-s + (−6.10 − 14.7i)11-s + (−2.82 + 6.80i)13-s − 20.3·15-s − 3.67i·17-s + (1.65 + 0.686i)19-s + (−11.7 + 4.86i)21-s + (−8.31 + 8.31i)23-s + (−31.2 + 31.2i)25-s + (−27.1 + 11.2i)27-s + (38.8 + 16.0i)29-s + 4.11i·31-s + ⋯
L(s)  = 1  + (−0.312 + 0.753i)3-s + (0.636 + 1.53i)5-s + (0.524 + 0.524i)7-s + (0.236 + 0.236i)9-s + (−0.554 − 1.33i)11-s + (−0.216 + 0.523i)13-s − 1.35·15-s − 0.215i·17-s + (0.0872 + 0.0361i)19-s + (−0.559 + 0.231i)21-s + (−0.361 + 0.361i)23-s + (−1.24 + 1.24i)25-s + (−1.00 + 0.416i)27-s + (1.33 + 0.555i)29-s + 0.132i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.591 - 0.806i$
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: \((2,\ 256,\ (\ :1),\ -0.591 - 0.806i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.694970 + 1.37177i\)
\(L(\frac12)\) \(\approx\) \(0.694970 + 1.37177i\)
\(L(2)\) 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 \( 1 + (0.936 - 2.26i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-3.18 - 7.68i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-3.67 - 3.67i)T + 49iT^{2} \)
11 \( 1 + (6.10 + 14.7i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (2.82 - 6.80i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 3.67iT - 289T^{2} \)
19 \( 1 + (-1.65 - 0.686i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (8.31 - 8.31i)T - 529iT^{2} \)
29 \( 1 + (-38.8 - 16.0i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 4.11iT - 961T^{2} \)
37 \( 1 + (19.8 + 47.9i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-21.1 - 21.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (-0.102 - 0.247i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 39.3T + 2.20e3T^{2} \)
53 \( 1 + (22.6 - 9.36i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-101. + 41.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-14.0 - 5.81i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (3.67 - 8.87i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-75.7 - 75.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (29.0 + 29.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 2.76T + 6.24e3T^{2} \)
83 \( 1 + (-79.1 - 32.8i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-72.4 + 72.4i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.0T + 9.40e3T^{2} \)
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

−11.70567622358942437433200533113, −10.98620319637252769762296624217, −10.40867210704674740442923317362, −9.523145426299138217181247698401, −8.265287658933095286891047078488, −7.04366451760931354500623577348, −5.95827882417906860142050821155, −5.06591383610526653948088504217, −3.48153822010776398844596142646, −2.27072739672886392908209050563, 0.860385025402268652783347766707, 1.98533202672042833282218644836, 4.41103792412156553755738538535, 5.15803828919762458134984703816, 6.41429534189964750597893991398, 7.58215510798685378472081074034, 8.379454896583810251009600976866, 9.652668916089991561493991631549, 10.29488502915497700493856318239, 11.86684408918236580754454671120

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