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Properties

Label 2-2e8-32.27-c2-0-7
Degree $2$
Conductor $256$
Sign $0.990 - 0.136i$
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  + (0.374 − 0.155i)3-s + (7.60 + 3.15i)5-s + (6.84 − 6.84i)7-s + (−6.24 + 6.24i)9-s + (2.23 + 0.927i)11-s + (−1.40 + 0.583i)13-s + 3.34·15-s − 2.67i·17-s + (−5.38 − 13.0i)19-s + (1.50 − 3.62i)21-s + (18.8 + 18.8i)23-s + (30.2 + 30.2i)25-s + (−2.77 + 6.68i)27-s + (10.0 + 24.2i)29-s − 47.5i·31-s + ⋯
L(s)  = 1  + (0.124 − 0.0517i)3-s + (1.52 + 0.630i)5-s + (0.977 − 0.977i)7-s + (−0.694 + 0.694i)9-s + (0.203 + 0.0842i)11-s + (−0.108 + 0.0449i)13-s + 0.222·15-s − 0.157i·17-s + (−0.283 − 0.684i)19-s + (0.0715 − 0.172i)21-s + (0.819 + 0.819i)23-s + (1.21 + 1.21i)25-s + (−0.102 + 0.247i)27-s + (0.345 + 0.834i)29-s − 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(6.97549\)
Root analytic conductor: \(2.64111\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1),\ 0.990 - 0.136i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.21267 + 0.151590i\)
\(L(\frac12)\) \(\approx\) \(2.21267 + 0.151590i\)
\(L(2)\) 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 \( 1 + (-0.374 + 0.155i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (-7.60 - 3.15i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (-6.84 + 6.84i)T - 49iT^{2} \)
11 \( 1 + (-2.23 - 0.927i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (1.40 - 0.583i)T + (119. - 119. i)T^{2} \)
17 \( 1 + 2.67iT - 289T^{2} \)
19 \( 1 + (5.38 + 13.0i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (-18.8 - 18.8i)T + 529iT^{2} \)
29 \( 1 + (-10.0 - 24.2i)T + (-594. + 594. i)T^{2} \)
31 \( 1 + 47.5iT - 961T^{2} \)
37 \( 1 + (-28.2 - 11.7i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (-6.93 + 6.93i)T - 1.68e3iT^{2} \)
43 \( 1 + (8.48 + 3.51i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 67.0T + 2.20e3T^{2} \)
53 \( 1 + (-10.5 + 25.3i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (27.9 - 67.4i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (31.5 + 76.2i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (90.1 - 37.3i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-1.98 + 1.98i)T - 5.04e3iT^{2} \)
73 \( 1 + (55.5 - 55.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 10.9T + 6.24e3T^{2} \)
83 \( 1 + (34.1 + 82.5i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (16.1 + 16.1i)T + 7.92e3iT^{2} \)
97 \( 1 + 62.6T + 9.40e3T^{2} \)
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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

−11.42680402385825696152060901081, −10.89029953365556464347037450411, −9.998738119948198683913498887864, −9.029361376586098145899555667504, −7.79547077822360727408148686239, −6.84883854369389047304356307865, −5.67804885812221586216802402014, −4.67076612482717816464709034386, −2.83458474831566858024092601140, −1.60758207338410386767884676980, 1.50268052888891775318209366466, 2.73958101981583287060327413826, 4.73734038510639163712208460406, 5.67133713890385781081886354127, 6.40420895727696376506956342855, 8.273351663517332738206616517663, 8.875919923454937128322408078537, 9.649115832780562940740928199358, 10.78128988869071897414294039026, 11.93443885441449478135651566285

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