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Properties

Label 2-2e8-32.19-c2-0-2
Degree $2$
Conductor $256$
Sign $0.971 - 0.238i$
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  + (−3.70 − 1.53i)3-s + (−7.20 + 2.98i)5-s + (−4.26 − 4.26i)7-s + (4.99 + 4.99i)9-s + (6.19 − 2.56i)11-s + (8.05 + 3.33i)13-s + 31.2·15-s + 24.5i·17-s + (4.96 − 11.9i)19-s + (9.24 + 22.3i)21-s + (9.72 − 9.72i)23-s + (25.2 − 25.2i)25-s + (2.97 + 7.17i)27-s + (5.86 − 14.1i)29-s + 17.5i·31-s + ⋯
L(s)  = 1  + (−1.23 − 0.511i)3-s + (−1.44 + 0.596i)5-s + (−0.608 − 0.608i)7-s + (0.554 + 0.554i)9-s + (0.563 − 0.233i)11-s + (0.619 + 0.256i)13-s + 2.08·15-s + 1.44i·17-s + (0.261 − 0.630i)19-s + (0.440 + 1.06i)21-s + (0.422 − 0.422i)23-s + (1.01 − 1.01i)25-s + (0.110 + 0.265i)27-s + (0.202 − 0.488i)29-s + 0.565i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.971 - 0.238i$
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: \((2,\ 256,\ (\ :1),\ 0.971 - 0.238i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.600514 + 0.0728072i\)
\(L(\frac12)\) \(\approx\) \(0.600514 + 0.0728072i\)
\(L(2)\) 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 \( 1 + (3.70 + 1.53i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (7.20 - 2.98i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (4.26 + 4.26i)T + 49iT^{2} \)
11 \( 1 + (-6.19 + 2.56i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-8.05 - 3.33i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 24.5iT - 289T^{2} \)
19 \( 1 + (-4.96 + 11.9i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-9.72 + 9.72i)T - 529iT^{2} \)
29 \( 1 + (-5.86 + 14.1i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 17.5iT - 961T^{2} \)
37 \( 1 + (-36.0 + 14.9i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-10.9 - 10.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (22.4 - 9.27i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 27.0T + 2.20e3T^{2} \)
53 \( 1 + (-34.0 - 82.1i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-27.8 - 67.2i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-6.37 + 15.3i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (99.2 + 41.0i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-2.55 - 2.55i)T + 5.04e3iT^{2} \)
73 \( 1 + (-30.7 - 30.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 90.6T + 6.24e3T^{2} \)
83 \( 1 + (39.3 - 94.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-109. + 109. i)T - 7.92e3iT^{2} \)
97 \( 1 - 63.7T + 9.40e3T^{2} \)
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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

−11.69079946287481680484129975316, −11.08605239390473241450230269705, −10.39554443570723142894326006052, −8.781742549745946580049671833815, −7.59973277211880835157496771362, −6.72547964216292987213039464713, −6.09984413336333580910272385686, −4.38902764086316569070481953260, −3.41558430014859479641040726200, −0.830601923349366206430936995445, 0.58133259928144996162506747743, 3.40709387656111101316836131507, 4.55888593530731225103319446655, 5.46435484885857402417206720793, 6.61184532258979735004322887547, 7.81147326576824920200810704827, 8.953495144743230564225023669095, 9.892651142023056102035760401026, 11.17562513676483075139228820877, 11.72280795423999379261302926064

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