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Properties

Label 2-2e8-32.19-c2-0-11
Degree $2$
Conductor $256$
Sign $-0.619 + 0.784i$
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  + (1.37 + 0.568i)3-s + (−2.28 + 0.948i)5-s + (−6.37 − 6.37i)7-s + (−4.80 − 4.80i)9-s + (1.79 − 0.744i)11-s + (−16.7 − 6.91i)13-s − 3.68·15-s + 6.19i·17-s + (8.50 − 20.5i)19-s + (−5.12 − 12.3i)21-s + (23.6 − 23.6i)23-s + (−13.3 + 13.3i)25-s + (−8.98 − 21.6i)27-s + (−14.5 + 35.1i)29-s + 14.1i·31-s + ⋯
L(s)  = 1  + (0.457 + 0.189i)3-s + (−0.457 + 0.189i)5-s + (−0.911 − 0.911i)7-s + (−0.533 − 0.533i)9-s + (0.163 − 0.0676i)11-s + (−1.28 − 0.532i)13-s − 0.245·15-s + 0.364i·17-s + (0.447 − 1.08i)19-s + (−0.244 − 0.589i)21-s + (1.02 − 1.02i)23-s + (−0.533 + 0.533i)25-s + (−0.332 − 0.802i)27-s + (−0.502 + 1.21i)29-s + 0.456i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.619 + 0.784i$
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: \((2,\ 256,\ (\ :1),\ -0.619 + 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.318128 - 0.656533i\)
\(L(\frac12)\) \(\approx\) \(0.318128 - 0.656533i\)
\(L(2)\) 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 \( 1 + (-1.37 - 0.568i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (2.28 - 0.948i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (6.37 + 6.37i)T + 49iT^{2} \)
11 \( 1 + (-1.79 + 0.744i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (16.7 + 6.91i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 6.19iT - 289T^{2} \)
19 \( 1 + (-8.50 + 20.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-23.6 + 23.6i)T - 529iT^{2} \)
29 \( 1 + (14.5 - 35.1i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 14.1iT - 961T^{2} \)
37 \( 1 + (-30.0 + 12.4i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (56.9 + 56.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (54.5 - 22.5i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 34.8T + 2.20e3T^{2} \)
53 \( 1 + (3.92 + 9.48i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (9.41 + 22.7i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (3.00 - 7.25i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-55.9 - 23.1i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-6.27 - 6.27i)T + 5.04e3iT^{2} \)
73 \( 1 + (-66.4 - 66.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.8T + 6.24e3T^{2} \)
83 \( 1 + (-1.23 + 2.97i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-36.7 + 36.7i)T - 7.92e3iT^{2} \)
97 \( 1 - 90.0T + 9.40e3T^{2} \)
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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

−11.40765567206545470145119854224, −10.40560047981088978644988570632, −9.538821746779472688912338964680, −8.645287107860587596775098570485, −7.34689021628589407881857256462, −6.71370959379601269665437537750, −5.13283267590536174162523383800, −3.71591133126862908294004832041, −2.89172844904051360267511107895, −0.33579563212769489900865639851, 2.24050984699871738964875962325, 3.37415389878317368742040583690, 4.94888881115854546481040384303, 6.07019984076787929675412264311, 7.37712410962215606366316534070, 8.198007498332671611204002798931, 9.330105710239702793720488662068, 9.889814399015463805516954567615, 11.56578275593180860223032990894, 11.98066981997709337430609443722

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